% COMMAND-LINE: --full-saturate-quant --ho-elim
% EXPECT: % SZS status Unsatisfiable for partial_app_interpreted_SWW474^2

%------------------------------------------------------------------------------
% File     : SWW474^2 : TPTP v7.2.0. Released v5.3.0.
% Domain   : Software Verification
% Problem  : Hoare's Logic with Procedures line 440, 500 axioms selected
% Version  : Especial.
% English  :

% Refs     : [BN10]  Boehme & Nipkow (2010), Sledgehammer: Judgement Day
%          : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source   : [Bla11]
% Names    : hoare_500_thf_l440 [Bla11]

% Status   : Theorem
% Rating   : 0.56 v7.2.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v6.1.0, 0.71 v5.5.0, 0.67 v5.4.0, 0.80 v5.3.0
% Syntax   : Number of formulae    :  833 (   2 unit; 110 type;   0 defn)
%            Number of atoms       : 8029 ( 594 equality;4565 variable)
%            Maximal formula depth :   19 (   8 average)
%            Number of connectives : 6344 ( 226   ~;  35   |; 108   &;5021   @)
%                                         ( 116 <=>; 838  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2404 (2404   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  118 ( 110   :;   0   =)
%            Number of variables   : 2030 (  12 sgn;1851   !;  48   ?; 131   ^)
%                                         (2030   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-08-09 19:43:11
%------------------------------------------------------------------------------
%----Should-be-implicit typings (7)
thf(ty_ty_tc__Com__Ocom,type,(
    com: $tType )).

thf(ty_ty_tc__Com__Opname,type,(
    pname: $tType )).

thf(ty_ty_tc__Com__Ostate,type,(
    state: $tType )).

thf(ty_ty_tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,(
    hoare_1167836817_state: $tType )).

thf(ty_ty_tc__Option__Ooption_Itc__Com__Ocom_J,type,(
    option_com: $tType )).

thf(ty_ty_tc__Option__Ooption_Itc__Com__Opname_J,type,(
    option_pname: $tType )).

thf(ty_ty_tc__Option__Ooption_Itc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Co,type,(
    option1574264306_state: $tType )).

%----Explicit typings (103)
thf(sy_c_Com_OWT,type,(
    wt: com > $o )).

thf(sy_c_Com_OWT__bodies,type,(
    wT_bodies: $o )).

thf(sy_c_Com_Obody,type,(
    body: pname > option_com )).

thf(sy_c_Com_Ocom_OBODY,type,(
    body_1: pname > com )).

thf(sy_c_Com_Ocom_OSKIP,type,(
    skip: com )).

thf(sy_c_Com_Ocom_OSemi,type,(
    semi: com > com > com )).

thf(sy_c_Com_Ocom_OWhile,type,(
    while: ( state > $o ) > com > com )).

thf(sy_c_Finite__Set_Ofinite_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_J,type,(
    finite1066544169me_o_o: ( ( ( pname > $o ) > $o ) > $o ) > $o )).

thf(sy_c_Finite__Set_Ofinite_000_062_I_062_Itc__Hoare____Mirabelle____srushsumbx__Ot,type,(
    finite33115244te_o_o: ( ( ( hoare_1167836817_state > $o ) > $o ) > $o ) > $o )).

thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Com__Opname_M_Eo_J,type,(
    finite297249702name_o: ( ( pname > $o ) > $o ) > $o )).

thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_,type,(
    finite1380128977tate_o: ( ( hoare_1167836817_state > $o ) > $o ) > $o )).

thf(sy_c_Finite__Set_Ofinite_000tc__Com__Opname,type,(
    finite_finite_pname: ( pname > $o ) > $o )).

thf(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__C,type,(
    finite1084549118_state: ( hoare_1167836817_state > $o ) > $o )).

thf(sy_c_Finite__Set_Ofolding__one_000_062_Itc__Com__Opname_M_Eo_J,type,(
    finite349908348name_o: ( ( pname > $o ) > ( pname > $o ) > pname > $o ) > ( ( ( pname > $o ) > $o ) > pname > $o ) > $o )).

thf(sy_c_Finite__Set_Ofolding__one_000_062_Itc__Hoare____Mirabelle____srushsumbx__Ot,type,(
    finite979047547tate_o: ( ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ) > ( ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o ) > $o )).

thf(sy_c_Finite__Set_Ofolding__one_000tc__Com__Opname,type,(
    finite1282449217_pname: ( pname > pname > pname ) > ( ( pname > $o ) > pname ) > $o )).

thf(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____srushsumbx__Otriple_,type,(
    finite1074406356_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state ) > $o )).

thf(sy_c_Finite__Set_Ofolding__one__idem_000_062_Itc__Com__Opname_M_Eo_J,type,(
    finite697516351name_o: ( ( pname > $o ) > ( pname > $o ) > pname > $o ) > ( ( ( pname > $o ) > $o ) > pname > $o ) > $o )).

thf(sy_c_Finite__Set_Ofolding__one__idem_000_062_Itc__Hoare____Mirabelle____srushsum,type,(
    finite671847800tate_o: ( ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ) > ( ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o ) > $o )).

thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Com__Opname,type,(
    finite89670078_pname: ( pname > pname > pname ) > ( ( pname > $o ) > pname ) > $o )).

thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____srushsumbx__Ot,type,(
    finite806517911_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state ) > $o )).

thf(sy_c_Groups_Ominus__class_Ominus_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_J,type,(
    minus_1480864103me_o_o: ( ( pname > $o ) > $o ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o )).

thf(sy_c_Groups_Ominus__class_Ominus_000_062_I_062_Itc__Hoare____Mirabelle____srushs,type,(
    minus_1708687022te_o_o: ( ( hoare_1167836817_state > $o ) > $o ) > ( ( hoare_1167836817_state > $o ) > $o ) > ( hoare_1167836817_state > $o ) > $o )).

thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Com__Opname_M_Eo_J,type,(
    minus_minus_pname_o: ( pname > $o ) > ( pname > $o ) > pname > $o )).

thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Hoare____Mirabelle____srushsumbx__,type,(
    minus_2107060239tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o )).

thf(sy_c_HOL_OThe_000tc__Com__Opname,type,(
    the_pname: ( pname > $o ) > pname )).

thf(sy_c_HOL_OThe_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_,type,(
    the_Ho310147232_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state )).

thf(sy_c_Hoare__Mirabelle__srushsumbx_OMGT,type,(
    hoare_Mirabelle_MGT: com > hoare_1167836817_state )).

thf(sy_c_Hoare__Mirabelle__srushsumbx_Ohoare__derivs_000tc__Com__Ostate,type,(
    hoare_123228589_state: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o )).

thf(sy_c_Hoare__Mirabelle__srushsumbx_Ohoare__valids_000tc__Com__Ostate,type,(
    hoare_529639851_state: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o )).

thf(sy_c_Hoare__Mirabelle__srushsumbx_Ostate__not__singleton,type,(
    hoare_1201148605gleton: $o )).

thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple_Otriple_000tc__Com__Ostate,type,(
    hoare_908217195_state: ( state > state > $o ) > com > ( state > state > $o ) > hoare_1167836817_state )).

thf(sy_c_If_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,(
    if_Hoa833675553_state: $o > hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state )).

thf(sy_c_If_000tc__Option__Ooption_Itc__Com__Ocom_J,type,(
    if_option_com: $o > option_com > option_com > option_com )).

thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_I_062_Itc__Com__Opname_M_Eo_,type,(
    semila2013987940me_o_o: ( ( pname > $o ) > $o ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o )).

thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_I_062_Itc__Hoare____Mirabell,type,(
    semila1758709489te_o_o: ( ( hoare_1167836817_state > $o ) > $o ) > ( ( hoare_1167836817_state > $o ) > $o ) > ( hoare_1167836817_state > $o ) > $o )).

thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_Itc__Com__Opname_M_Eo_J,type,(
    semila1673364395name_o: ( pname > $o ) > ( pname > $o ) > pname > $o )).

thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_Itc__Hoare____Mirabelle____s,type,(
    semila179895820tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o )).

thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_I_062_Itc__Com__Opname_M_Eo_,type,(
    semila181081674me_o_o: ( ( pname > $o ) > $o ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o )).

thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_I_062_Itc__Hoare____Mirabell,type,(
    semila866907787te_o_o: ( ( hoare_1167836817_state > $o ) > $o ) > ( ( hoare_1167836817_state > $o ) > $o ) > ( hoare_1167836817_state > $o ) > $o )).

thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Com__Opname_M_Eo_J,type,(
    semila1780557381name_o: ( pname > $o ) > ( pname > $o ) > pname > $o )).

thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Hoare____Mirabelle____s,type,(
    semila1172322802tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o )).

thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_Eo,type,(
    semila10642723_sup_o: $o > $o > $o )).

thf(sy_c_Map_Odom_000tc__Com__Opname_000tc__Com__Ocom,type,(
    dom_pname_com: ( pname > option_com ) > pname > $o )).

thf(sy_c_Natural_Oevalc,type,(
    evalc: com > state > state > $o )).

thf(sy_c_Option_Ooption_OSome_000tc__Com__Ocom,type,(
    some_com: com > option_com )).

thf(sy_c_Option_Ooption_OSome_000tc__Com__Opname,type,(
    some_pname: pname > option_pname )).

thf(sy_c_Option_Ooption_OSome_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__,type,(
    some_H1433514562_state: hoare_1167836817_state > option1574264306_state )).

thf(sy_c_Option_Oset_000tc__Com__Ocom,type,(
    set_com: option_com > com > $o )).

thf(sy_c_Option_Oset_000tc__Com__Opname,type,(
    set_pname: option_pname > pname > $o )).

thf(sy_c_Option_Oset_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Osta,type,(
    set_Ho2131684873_state: option1574264306_state > hoare_1167836817_state > $o )).

thf(sy_c_Option_Othe_000tc__Com__Ocom,type,(
    the_com: option_com > com )).

thf(sy_c_Orderings_Obot__class_Obot_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_J,type,(
    bot_bot_pname_o_o: ( pname > $o ) > $o )).

thf(sy_c_Orderings_Obot__class_Obot_000_062_I_062_Itc__Hoare____Mirabelle____srushsu,type,(
    bot_bo691907561te_o_o: ( hoare_1167836817_state > $o ) > $o )).

thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Com__Ocom_M_Eo_J,type,(
    bot_bot_com_o: com > $o )).

thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Com__Opname_M_Eo_J,type,(
    bot_bot_pname_o: pname > $o )).

thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Hoare____Mirabelle____srushsumbx__O,type,(
    bot_bo70021908tate_o: hoare_1167836817_state > $o )).

thf(sy_c_Orderings_Obot__class_Obot_000_Eo,type,(
    bot_bot_o: $o )).

thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_,type,(
    ord_le1205211808me_o_o: ( ( pname > $o ) > $o ) > ( ( pname > $o ) > $o ) > $o )).

thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_Itc__Hoare____Mirabelle____sr,type,(
    ord_le741939125te_o_o: ( ( hoare_1167836817_state > $o ) > $o ) > ( ( hoare_1167836817_state > $o ) > $o ) > $o )).

thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Com__Opname_M_Eo_J,type,(
    ord_less_eq_pname_o: ( pname > $o ) > ( pname > $o ) > $o )).

thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Hoare____Mirabelle____srushsum,type,(
    ord_le827224136tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o )).

thf(sy_c_Orderings_Oord__class_Oless__eq_000_Eo,type,(
    ord_less_eq_o: $o > $o > $o )).

thf(sy_c_Set_OCollect_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_J,type,(
    collect_pname_o_o: ( ( ( pname > $o ) > $o ) > $o ) > ( ( pname > $o ) > $o ) > $o )).

thf(sy_c_Set_OCollect_000_062_I_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_I,type,(
    collec1218656682te_o_o: ( ( ( hoare_1167836817_state > $o ) > $o ) > $o ) > ( ( hoare_1167836817_state > $o ) > $o ) > $o )).

thf(sy_c_Set_OCollect_000_062_Itc__Com__Opname_M_Eo_J,type,(
    collect_pname_o: ( ( pname > $o ) > $o ) > ( pname > $o ) > $o )).

thf(sy_c_Set_OCollect_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Co,type,(
    collec269976083tate_o: ( ( hoare_1167836817_state > $o ) > $o ) > ( hoare_1167836817_state > $o ) > $o )).

thf(sy_c_Set_OCollect_000tc__Com__Opname,type,(
    collect_pname: ( pname > $o ) > pname > $o )).

thf(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ost,type,(
    collec1027672124_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o )).

thf(sy_c_Set_Oimage_000_062_Itc__Com__Opname_M_Eo_J_000_062_Itc__Com__Opname_M_Eo_J,type,(
    image_1085733413name_o: ( ( pname > $o ) > pname > $o ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o )).

thf(sy_c_Set_Oimage_000_062_Itc__Com__Opname_M_Eo_J_000tc__Com__Opname,type,(
    image_pname_o_pname: ( ( pname > $o ) > pname ) > ( ( pname > $o ) > $o ) > pname > $o )).

thf(sy_c_Set_Oimage_000_062_Itc__Com__Opname_M_Eo_J_000tc__Hoare____Mirabelle____sru,type,(
    image_1381916541_state: ( ( pname > $o ) > hoare_1167836817_state ) > ( ( pname > $o ) > $o ) > hoare_1167836817_state > $o )).

thf(sy_c_Set_Oimage_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com_,type,(
    image_1488525317tate_o: ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ) > ( ( hoare_1167836817_state > $o ) > $o ) > ( hoare_1167836817_state > $o ) > $o )).

thf(sy_c_Set_Oimage_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__001,type,(
    image_980295115_pname: ( ( hoare_1167836817_state > $o ) > pname ) > ( ( hoare_1167836817_state > $o ) > $o ) > pname > $o )).

thf(sy_c_Set_Oimage_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__002,type,(
    image_635813834_state: ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state ) > ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o )).

thf(sy_c_Set_Oimage_000tc__Com__Opname_000_062_Itc__Com__Opname_M_Eo_J,type,(
    image_pname_pname_o: ( pname > pname > $o ) > ( pname > $o ) > ( pname > $o ) > $o )).

thf(sy_c_Set_Oimage_000tc__Com__Opname_000_062_Itc__Hoare____Mirabelle____srushsumbx,type,(
    image_475339327tate_o: ( pname > hoare_1167836817_state > $o ) > ( pname > $o ) > ( hoare_1167836817_state > $o ) > $o )).

thf(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Com__Opname,type,(
    image_pname_pname: ( pname > pname ) > ( pname > $o ) > pname > $o )).

thf(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Hoare____Mirabelle____srushsumbx__Otri,type,(
    image_575578384_state: ( pname > hoare_1167836817_state ) > ( pname > $o ) > hoare_1167836817_state > $o )).

thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostat,type,(
    image_2066861949name_o: ( hoare_1167836817_state > pname > $o ) > ( hoare_1167836817_state > $o ) > ( pname > $o ) > $o )).

thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostat_003,type,(
    image_1745649338tate_o: ( hoare_1167836817_state > hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o )).

thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostat_004,type,(
    image_8178176_pname: ( hoare_1167836817_state > pname ) > ( hoare_1167836817_state > $o ) > pname > $o )).

thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostat_005,type,(
    image_31595733_state: ( hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o )).

thf(sy_c_Set_Oinsert_000_062_Itc__Com__Opname_M_Eo_J,type,(
    insert_pname_o: ( pname > $o ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o )).

thf(sy_c_Set_Oinsert_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,type,(
    insert999278200tate_o: ( hoare_1167836817_state > $o ) > ( ( hoare_1167836817_state > $o ) > $o ) > ( hoare_1167836817_state > $o ) > $o )).

thf(sy_c_Set_Oinsert_000tc__Com__Ocom,type,(
    insert_com: com > ( com > $o ) > com > $o )).

thf(sy_c_Set_Oinsert_000tc__Com__Opname,type,(
    insert_pname: pname > ( pname > $o ) > pname > $o )).

thf(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Osta,type,(
    insert2134838167_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o )).

thf(sy_c_Set_Othe__elem_000tc__Com__Opname,type,(
    the_elem_pname: ( pname > $o ) > pname )).

thf(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__O,type,(
    the_el323660082_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state )).

thf(sy_c_fequal_000_062_Itc__Com__Opname_M_Eo_J,type,(
    fequal_pname_o: ( pname > $o ) > ( pname > $o ) > $o )).

thf(sy_c_fequal_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ost,type,(
    fequal1486222077tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o )).

thf(sy_c_fequal_000tc__Com__Opname,type,(
    fequal_pname: pname > pname > $o )).

thf(sy_c_fequal_000tc__Com__Ostate,type,(
    fequal_state: state > state > $o )).

thf(sy_c_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,(
    fequal1831255762_state: hoare_1167836817_state > hoare_1167836817_state > $o )).

thf(sy_c_member_000_062_Itc__Com__Opname_M_Eo_J,type,(
    member_pname_o: ( pname > $o ) > ( ( pname > $o ) > $o ) > $o )).

thf(sy_c_member_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ost,type,(
    member864234961tate_o: ( hoare_1167836817_state > $o ) > ( ( hoare_1167836817_state > $o ) > $o ) > $o )).

thf(sy_c_member_000tc__Com__Ocom,type,(
    member_com: com > ( com > $o ) > $o )).

thf(sy_c_member_000tc__Com__Opname,type,(
    member_pname: pname > ( pname > $o ) > $o )).

thf(sy_c_member_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,(
    member2058392318_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > $o )).

thf(sy_v_Fa,type,(
    fa: hoare_1167836817_state > $o )).

thf(sy_v_pn,type,(
    pn: pname )).

thf(sy_v_y,type,(
    y: com )).

%----Relevant facts (699)
thf(fact_0_empty,axiom,(
    ! [G_3: hoare_1167836817_state > $o] :
      ( hoare_123228589_state @ G_3 @ bot_bo70021908tate_o ) )).

thf(fact_1_asm,axiom,(
    ! [Ts_7: hoare_1167836817_state > $o,G_35: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ Ts_7 @ G_35 )
     => ( hoare_123228589_state @ G_35 @ Ts_7 ) ) )).

thf(fact_2_weaken,axiom,(
    ! [Ts_6: hoare_1167836817_state > $o,G_34: hoare_1167836817_state > $o,Ts_5: hoare_1167836817_state > $o] :
      ( ( hoare_123228589_state @ G_34 @ Ts_5 )
     => ( ( ord_le827224136tate_o @ Ts_6 @ Ts_5 )
       => ( hoare_123228589_state @ G_34 @ Ts_6 ) ) ) )).

thf(fact_3_thin,axiom,(
    ! [G_33: hoare_1167836817_state > $o,G_32: hoare_1167836817_state > $o,Ts_4: hoare_1167836817_state > $o] :
      ( ( hoare_123228589_state @ G_32 @ Ts_4 )
     => ( ( ord_le827224136tate_o @ G_32 @ G_33 )
       => ( hoare_123228589_state @ G_33 @ Ts_4 ) ) ) )).

thf(fact_4_cut,axiom,(
    ! [G_31: hoare_1167836817_state > $o,G_30: hoare_1167836817_state > $o,Ts_3: hoare_1167836817_state > $o] :
      ( ( hoare_123228589_state @ G_30 @ Ts_3 )
     => ( ( hoare_123228589_state @ G_31 @ G_30 )
       => ( hoare_123228589_state @ G_31 @ Ts_3 ) ) ) )).

thf(fact_5_hoare__derivs_Oinsert,axiom,(
    ! [Ts_2: hoare_1167836817_state > $o,G_29: hoare_1167836817_state > $o,T_3: hoare_1167836817_state] :
      ( ( hoare_123228589_state @ G_29 @ ( insert2134838167_state @ T_3 @ bot_bo70021908tate_o ) )
     => ( ( hoare_123228589_state @ G_29 @ Ts_2 )
       => ( hoare_123228589_state @ G_29 @ ( insert2134838167_state @ T_3 @ Ts_2 ) ) ) ) )).

thf(fact_6_derivs__insertD,axiom,(
    ! [G_28: hoare_1167836817_state > $o,T_2: hoare_1167836817_state,Ts_1: hoare_1167836817_state > $o] :
      ( ( hoare_123228589_state @ G_28 @ ( insert2134838167_state @ T_2 @ Ts_1 ) )
     => ( ( hoare_123228589_state @ G_28 @ ( insert2134838167_state @ T_2 @ bot_bo70021908tate_o ) )
        & ( hoare_123228589_state @ G_28 @ Ts_1 ) ) ) )).

thf(fact_7_MGT__BodyN,axiom,(
    ! [Pn_1: pname,G_3: hoare_1167836817_state > $o] :
      ( ( hoare_123228589_state @ ( insert2134838167_state @ ( hoare_Mirabelle_MGT @ ( body_1 @ Pn_1 ) ) @ G_3 ) @ ( insert2134838167_state @ ( hoare_Mirabelle_MGT @ ( the_com @ ( body @ Pn_1 ) ) ) @ bot_bo70021908tate_o ) )
     => ( hoare_123228589_state @ G_3 @ ( insert2134838167_state @ ( hoare_Mirabelle_MGT @ ( body_1 @ Pn_1 ) ) @ bot_bo70021908tate_o ) ) ) )).

thf(fact_8_finite__Collect__subsets,axiom,(
    ! [A_201: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ A_201 )
     => ( finite1066544169me_o_o
        @ ( collect_pname_o_o
          @ ^ [B_43: ( pname > $o ) > $o] :
              ( ord_le1205211808me_o_o @ B_43 @ A_201 ) ) ) ) )).

thf(fact_9_finite__Collect__subsets,axiom,(
    ! [A_201: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ A_201 )
     => ( finite33115244te_o_o
        @ ( collec1218656682te_o_o
          @ ^ [B_43: ( hoare_1167836817_state > $o ) > $o] :
              ( ord_le741939125te_o_o @ B_43 @ A_201 ) ) ) ) )).

thf(fact_10_finite__Collect__subsets,axiom,(
    ! [A_201: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ A_201 )
     => ( finite1380128977tate_o
        @ ( collec269976083tate_o
          @ ^ [B_43: hoare_1167836817_state > $o] :
              ( ord_le827224136tate_o @ B_43 @ A_201 ) ) ) ) )).

thf(fact_11_finite__Collect__subsets,axiom,(
    ! [A_201: pname > $o] :
      ( ( finite_finite_pname @ A_201 )
     => ( finite297249702name_o
        @ ( collect_pname_o
          @ ^ [B_43: pname > $o] :
              ( ord_less_eq_pname_o @ B_43 @ A_201 ) ) ) ) )).

thf(fact_12_finite__imageI,axiom,(
    ! [H_1: hoare_1167836817_state > pname > $o,F_61: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ F_61 )
     => ( finite297249702name_o @ ( image_2066861949name_o @ H_1 @ F_61 ) ) ) )).

thf(fact_13_finite__imageI,axiom,(
    ! [H_1: hoare_1167836817_state > hoare_1167836817_state > $o,F_61: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ F_61 )
     => ( finite1380128977tate_o @ ( image_1745649338tate_o @ H_1 @ F_61 ) ) ) )).

thf(fact_14_finite__imageI,axiom,(
    ! [H_1: pname > pname > $o,F_61: pname > $o] :
      ( ( finite_finite_pname @ F_61 )
     => ( finite297249702name_o @ ( image_pname_pname_o @ H_1 @ F_61 ) ) ) )).

thf(fact_15_finite__imageI,axiom,(
    ! [H_1: pname > hoare_1167836817_state > $o,F_61: pname > $o] :
      ( ( finite_finite_pname @ F_61 )
     => ( finite1380128977tate_o @ ( image_475339327tate_o @ H_1 @ F_61 ) ) ) )).

thf(fact_16_finite__imageI,axiom,(
    ! [H_1: ( pname > $o ) > hoare_1167836817_state,F_61: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ F_61 )
     => ( finite1084549118_state @ ( image_1381916541_state @ H_1 @ F_61 ) ) ) )).

thf(fact_17_finite__imageI,axiom,(
    ! [H_1: ( hoare_1167836817_state > $o ) > hoare_1167836817_state,F_61: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ F_61 )
     => ( finite1084549118_state @ ( image_635813834_state @ H_1 @ F_61 ) ) ) )).

thf(fact_18_finite__imageI,axiom,(
    ! [H_1: ( pname > $o ) > pname,F_61: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ F_61 )
     => ( finite_finite_pname @ ( image_pname_o_pname @ H_1 @ F_61 ) ) ) )).

thf(fact_19_finite__imageI,axiom,(
    ! [H_1: ( hoare_1167836817_state > $o ) > pname,F_61: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ F_61 )
     => ( finite_finite_pname @ ( image_980295115_pname @ H_1 @ F_61 ) ) ) )).

thf(fact_20_finite__imageI,axiom,(
    ! [H_1: pname > hoare_1167836817_state,F_61: pname > $o] :
      ( ( finite_finite_pname @ F_61 )
     => ( finite1084549118_state @ ( image_575578384_state @ H_1 @ F_61 ) ) ) )).

thf(fact_21_empty__subsetI,axiom,(
    ! [A_200: pname > $o] :
      ( ord_less_eq_pname_o @ bot_bot_pname_o @ A_200 ) )).

thf(fact_22_empty__subsetI,axiom,(
    ! [A_200: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ bot_bo70021908tate_o @ A_200 ) )).

thf(fact_23_finite_OinsertI,axiom,(
    ! [A_199: pname > $o,A_198: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ A_198 )
     => ( finite297249702name_o @ ( insert_pname_o @ A_199 @ A_198 ) ) ) )).

thf(fact_24_finite_OinsertI,axiom,(
    ! [A_199: hoare_1167836817_state > $o,A_198: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ A_198 )
     => ( finite1380128977tate_o @ ( insert999278200tate_o @ A_199 @ A_198 ) ) ) )).

thf(fact_25_finite_OinsertI,axiom,(
    ! [A_199: pname,A_198: pname > $o] :
      ( ( finite_finite_pname @ A_198 )
     => ( finite_finite_pname @ ( insert_pname @ A_199 @ A_198 ) ) ) )).

thf(fact_26_finite_OinsertI,axiom,(
    ! [A_199: hoare_1167836817_state,A_198: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ A_198 )
     => ( finite1084549118_state @ ( insert2134838167_state @ A_199 @ A_198 ) ) ) )).

thf(fact_27_finite_OemptyI,axiom,
    ( finite297249702name_o @ bot_bot_pname_o_o )).

thf(fact_28_finite_OemptyI,axiom,
    ( finite1380128977tate_o @ bot_bo691907561te_o_o )).

thf(fact_29_finite_OemptyI,axiom,
    ( finite1084549118_state @ bot_bo70021908tate_o )).

thf(fact_30_finite_OemptyI,axiom,
    ( finite_finite_pname @ bot_bot_pname_o )).

thf(fact_31_finite__Collect__conjI,axiom,(
    ! [Q_24: ( pname > $o ) > $o,P_42: ( pname > $o ) > $o] :
      ( ( ( finite297249702name_o @ ( collect_pname_o @ P_42 ) )
        | ( finite297249702name_o @ ( collect_pname_o @ Q_24 ) ) )
     => ( finite297249702name_o
        @ ( collect_pname_o
          @ ^ [X_5: pname > $o] :
              ( (&) @ ( P_42 @ X_5 ) @ ( Q_24 @ X_5 ) ) ) ) ) )).

thf(fact_32_finite__Collect__conjI,axiom,(
    ! [Q_24: ( hoare_1167836817_state > $o ) > $o,P_42: ( hoare_1167836817_state > $o ) > $o] :
      ( ( ( finite1380128977tate_o @ ( collec269976083tate_o @ P_42 ) )
        | ( finite1380128977tate_o @ ( collec269976083tate_o @ Q_24 ) ) )
     => ( finite1380128977tate_o
        @ ( collec269976083tate_o
          @ ^ [X_5: hoare_1167836817_state > $o] :
              ( (&) @ ( P_42 @ X_5 ) @ ( Q_24 @ X_5 ) ) ) ) ) )).

thf(fact_33_finite__Collect__conjI,axiom,(
    ! [Q_24: hoare_1167836817_state > $o,P_42: hoare_1167836817_state > $o] :
      ( ( ( finite1084549118_state @ ( collec1027672124_state @ P_42 ) )
        | ( finite1084549118_state @ ( collec1027672124_state @ Q_24 ) ) )
     => ( finite1084549118_state
        @ ( collec1027672124_state
          @ ^ [X_5: hoare_1167836817_state] :
              ( (&) @ ( P_42 @ X_5 ) @ ( Q_24 @ X_5 ) ) ) ) ) )).

thf(fact_34_finite__Collect__conjI,axiom,(
    ! [Q_24: pname > $o,P_42: pname > $o] :
      ( ( ( finite_finite_pname @ ( collect_pname @ P_42 ) )
        | ( finite_finite_pname @ ( collect_pname @ Q_24 ) ) )
     => ( finite_finite_pname
        @ ( collect_pname
          @ ^ [X_5: pname] :
              ( (&) @ ( P_42 @ X_5 ) @ ( Q_24 @ X_5 ) ) ) ) ) )).

thf(fact_35_image__constant__conv,axiom,(
    ! [C_54: pname,A_197: hoare_1167836817_state > $o] :
      ( ( ( A_197 = bot_bo70021908tate_o )
       => ( ( image_8178176_pname
            @ ^ [X_5: hoare_1167836817_state] : C_54
            @ A_197 )
          = bot_bot_pname_o ) )
      & ( ( A_197 != bot_bo70021908tate_o )
       => ( ( image_8178176_pname
            @ ^ [X_5: hoare_1167836817_state] : C_54
            @ A_197 )
          = ( insert_pname @ C_54 @ bot_bot_pname_o ) ) ) ) )).

thf(fact_36_image__constant__conv,axiom,(
    ! [C_54: hoare_1167836817_state,A_197: pname > $o] :
      ( ( ( A_197 = bot_bot_pname_o )
       => ( ( image_575578384_state
            @ ^ [X_5: pname] : C_54
            @ A_197 )
          = bot_bo70021908tate_o ) )
      & ( ( A_197 != bot_bot_pname_o )
       => ( ( image_575578384_state
            @ ^ [X_5: pname] : C_54
            @ A_197 )
          = ( insert2134838167_state @ C_54 @ bot_bo70021908tate_o ) ) ) ) )).

thf(fact_37_image__constant,axiom,(
    ! [C_53: pname,X_101: hoare_1167836817_state,A_196: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ X_101 @ A_196 )
     => ( ( image_8178176_pname
          @ ^ [X_5: hoare_1167836817_state] : C_53
          @ A_196 )
        = ( insert_pname @ C_53 @ bot_bot_pname_o ) ) ) )).

thf(fact_38_image__constant,axiom,(
    ! [C_53: hoare_1167836817_state,X_101: pname,A_196: pname > $o] :
      ( ( member_pname @ X_101 @ A_196 )
     => ( ( image_575578384_state
          @ ^ [X_5: pname] : C_53
          @ A_196 )
        = ( insert2134838167_state @ C_53 @ bot_bo70021908tate_o ) ) ) )).

thf(fact_39_insert__dom,axiom,(
    ! [F_60: pname > option_com,X_100: pname,Y_49: com] :
      ( ( ( F_60 @ X_100 )
        = ( some_com @ Y_49 ) )
     => ( ( insert_pname @ X_100 @ ( dom_pname_com @ F_60 ) )
        = ( dom_pname_com @ F_60 ) ) ) )).

thf(fact_40_finite__surj,axiom,(
    ! [B_130: ( pname > $o ) > $o,F_59: hoare_1167836817_state > pname > $o,A_195: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ A_195 )
     => ( ( ord_le1205211808me_o_o @ B_130 @ ( image_2066861949name_o @ F_59 @ A_195 ) )
       => ( finite297249702name_o @ B_130 ) ) ) )).

thf(fact_41_finite__surj,axiom,(
    ! [B_130: ( hoare_1167836817_state > $o ) > $o,F_59: hoare_1167836817_state > hoare_1167836817_state > $o,A_195: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ A_195 )
     => ( ( ord_le741939125te_o_o @ B_130 @ ( image_1745649338tate_o @ F_59 @ A_195 ) )
       => ( finite1380128977tate_o @ B_130 ) ) ) )).

thf(fact_42_finite__surj,axiom,(
    ! [B_130: pname > $o,F_59: hoare_1167836817_state > pname,A_195: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ A_195 )
     => ( ( ord_less_eq_pname_o @ B_130 @ ( image_8178176_pname @ F_59 @ A_195 ) )
       => ( finite_finite_pname @ B_130 ) ) ) )).

thf(fact_43_finite__surj,axiom,(
    ! [B_130: ( pname > $o ) > $o,F_59: pname > pname > $o,A_195: pname > $o] :
      ( ( finite_finite_pname @ A_195 )
     => ( ( ord_le1205211808me_o_o @ B_130 @ ( image_pname_pname_o @ F_59 @ A_195 ) )
       => ( finite297249702name_o @ B_130 ) ) ) )).

thf(fact_44_finite__surj,axiom,(
    ! [B_130: ( hoare_1167836817_state > $o ) > $o,F_59: pname > hoare_1167836817_state > $o,A_195: pname > $o] :
      ( ( finite_finite_pname @ A_195 )
     => ( ( ord_le741939125te_o_o @ B_130 @ ( image_475339327tate_o @ F_59 @ A_195 ) )
       => ( finite1380128977tate_o @ B_130 ) ) ) )).

thf(fact_45_finite__surj,axiom,(
    ! [B_130: pname > $o,F_59: pname > pname,A_195: pname > $o] :
      ( ( finite_finite_pname @ A_195 )
     => ( ( ord_less_eq_pname_o @ B_130 @ ( image_pname_pname @ F_59 @ A_195 ) )
       => ( finite_finite_pname @ B_130 ) ) ) )).

thf(fact_46_finite__surj,axiom,(
    ! [B_130: hoare_1167836817_state > $o,F_59: ( pname > $o ) > hoare_1167836817_state,A_195: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ A_195 )
     => ( ( ord_le827224136tate_o @ B_130 @ ( image_1381916541_state @ F_59 @ A_195 ) )
       => ( finite1084549118_state @ B_130 ) ) ) )).

thf(fact_47_finite__surj,axiom,(
    ! [B_130: hoare_1167836817_state > $o,F_59: ( hoare_1167836817_state > $o ) > hoare_1167836817_state,A_195: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ A_195 )
     => ( ( ord_le827224136tate_o @ B_130 @ ( image_635813834_state @ F_59 @ A_195 ) )
       => ( finite1084549118_state @ B_130 ) ) ) )).

thf(fact_48_finite__surj,axiom,(
    ! [B_130: pname > $o,F_59: ( pname > $o ) > pname,A_195: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ A_195 )
     => ( ( ord_less_eq_pname_o @ B_130 @ ( image_pname_o_pname @ F_59 @ A_195 ) )
       => ( finite_finite_pname @ B_130 ) ) ) )).

thf(fact_49_finite__surj,axiom,(
    ! [B_130: pname > $o,F_59: ( hoare_1167836817_state > $o ) > pname,A_195: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ A_195 )
     => ( ( ord_less_eq_pname_o @ B_130 @ ( image_980295115_pname @ F_59 @ A_195 ) )
       => ( finite_finite_pname @ B_130 ) ) ) )).

thf(fact_50_finite__surj,axiom,(
    ! [B_130: hoare_1167836817_state > $o,F_59: pname > hoare_1167836817_state,A_195: pname > $o] :
      ( ( finite_finite_pname @ A_195 )
     => ( ( ord_le827224136tate_o @ B_130 @ ( image_575578384_state @ F_59 @ A_195 ) )
       => ( finite1084549118_state @ B_130 ) ) ) )).

thf(fact_51_subset__singletonD,axiom,(
    ! [A_194: pname > $o,X_99: pname] :
      ( ( ord_less_eq_pname_o @ A_194 @ ( insert_pname @ X_99 @ bot_bot_pname_o ) )
     => ( ( A_194 = bot_bot_pname_o )
        | ( A_194
          = ( insert_pname @ X_99 @ bot_bot_pname_o ) ) ) ) )).

thf(fact_52_subset__singletonD,axiom,(
    ! [A_194: hoare_1167836817_state > $o,X_99: hoare_1167836817_state] :
      ( ( ord_le827224136tate_o @ A_194 @ ( insert2134838167_state @ X_99 @ bot_bo70021908tate_o ) )
     => ( ( A_194 = bot_bo70021908tate_o )
        | ( A_194
          = ( insert2134838167_state @ X_99 @ bot_bo70021908tate_o ) ) ) ) )).

thf(fact_53_MGF,axiom,(
    ! [C_21: com] :
      ( hoare_1201148605gleton
     => ( wT_bodies
       => ( ( wt @ C_21 )
         => ( hoare_123228589_state @ bot_bo70021908tate_o @ ( insert2134838167_state @ ( hoare_Mirabelle_MGT @ C_21 ) @ bot_bo70021908tate_o ) ) ) ) ) )).

thf(fact_54_emptyE,axiom,(
    ! [A_193: pname] :
      ~ ( member_pname @ A_193 @ bot_bot_pname_o ) )).

thf(fact_55_emptyE,axiom,(
    ! [A_193: hoare_1167836817_state] :
      ~ ( member2058392318_state @ A_193 @ bot_bo70021908tate_o ) )).

thf(fact_56_insertCI,axiom,(
    ! [B_129: pname,A_192: pname,B_128: pname > $o] :
      ( ( ~ ( member_pname @ A_192 @ B_128 )
       => ( A_192 = B_129 ) )
     => ( member_pname @ A_192 @ ( insert_pname @ B_129 @ B_128 ) ) ) )).

thf(fact_57_insertCI,axiom,(
    ! [B_129: hoare_1167836817_state,A_192: hoare_1167836817_state,B_128: hoare_1167836817_state > $o] :
      ( ( ~ ( member2058392318_state @ A_192 @ B_128 )
       => ( A_192 = B_129 ) )
     => ( member2058392318_state @ A_192 @ ( insert2134838167_state @ B_129 @ B_128 ) ) ) )).

thf(fact_58_insertE,axiom,(
    ! [A_191: pname,B_127: pname,A_190: pname > $o] :
      ( ( member_pname @ A_191 @ ( insert_pname @ B_127 @ A_190 ) )
     => ( ( A_191 != B_127 )
       => ( member_pname @ A_191 @ A_190 ) ) ) )).

thf(fact_59_insertE,axiom,(
    ! [A_191: hoare_1167836817_state,B_127: hoare_1167836817_state,A_190: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ A_191 @ ( insert2134838167_state @ B_127 @ A_190 ) )
     => ( ( A_191 != B_127 )
       => ( member2058392318_state @ A_191 @ A_190 ) ) ) )).

thf(fact_60_equalityI,axiom,(
    ! [A_189: pname > $o,B_126: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_189 @ B_126 )
     => ( ( ord_less_eq_pname_o @ B_126 @ A_189 )
       => ( A_189 = B_126 ) ) ) )).

thf(fact_61_equalityI,axiom,(
    ! [A_189: hoare_1167836817_state > $o,B_126: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_189 @ B_126 )
     => ( ( ord_le827224136tate_o @ B_126 @ A_189 )
       => ( A_189 = B_126 ) ) ) )).

thf(fact_62_subsetD,axiom,(
    ! [C_52: pname,A_188: pname > $o,B_125: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_188 @ B_125 )
     => ( ( member_pname @ C_52 @ A_188 )
       => ( member_pname @ C_52 @ B_125 ) ) ) )).

thf(fact_63_subsetD,axiom,(
    ! [C_52: hoare_1167836817_state,A_188: hoare_1167836817_state > $o,B_125: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_188 @ B_125 )
     => ( ( member2058392318_state @ C_52 @ A_188 )
       => ( member2058392318_state @ C_52 @ B_125 ) ) ) )).

thf(fact_64_image__eqI,axiom,(
    ! [A_187: hoare_1167836817_state > $o,B_124: pname,F_58: hoare_1167836817_state > pname,X_98: hoare_1167836817_state] :
      ( ( B_124
        = ( F_58 @ X_98 ) )
     => ( ( member2058392318_state @ X_98 @ A_187 )
       => ( member_pname @ B_124 @ ( image_8178176_pname @ F_58 @ A_187 ) ) ) ) )).

thf(fact_65_image__eqI,axiom,(
    ! [A_187: pname > $o,B_124: hoare_1167836817_state,F_58: pname > hoare_1167836817_state,X_98: pname] :
      ( ( B_124
        = ( F_58 @ X_98 ) )
     => ( ( member_pname @ X_98 @ A_187 )
       => ( member2058392318_state @ B_124 @ ( image_575578384_state @ F_58 @ A_187 ) ) ) ) )).

thf(fact_66_equals0D,axiom,(
    ! [A_186: pname,A_185: pname > $o] :
      ( ( A_185 = bot_bot_pname_o )
     => ~ ( member_pname @ A_186 @ A_185 ) ) )).

thf(fact_67_equals0D,axiom,(
    ! [A_186: hoare_1167836817_state,A_185: hoare_1167836817_state > $o] :
      ( ( A_185 = bot_bo70021908tate_o )
     => ~ ( member2058392318_state @ A_186 @ A_185 ) ) )).

thf(fact_68_Collect__empty__eq,axiom,(
    ! [P_41: pname > $o] :
      ( ( ( collect_pname @ P_41 )
        = bot_bot_pname_o )
    <=> ! [X_5: pname] :
          ~ ( P_41 @ X_5 ) ) )).

thf(fact_69_Collect__empty__eq,axiom,(
    ! [P_41: ( pname > $o ) > $o] :
      ( ( ( collect_pname_o @ P_41 )
        = bot_bot_pname_o_o )
    <=> ! [X_5: pname > $o] :
          ~ ( P_41 @ X_5 ) ) )).

thf(fact_70_Collect__empty__eq,axiom,(
    ! [P_41: ( hoare_1167836817_state > $o ) > $o] :
      ( ( ( collec269976083tate_o @ P_41 )
        = bot_bo691907561te_o_o )
    <=> ! [X_5: hoare_1167836817_state > $o] :
          ~ ( P_41 @ X_5 ) ) )).

thf(fact_71_Collect__empty__eq,axiom,(
    ! [P_41: hoare_1167836817_state > $o] :
      ( ( ( collec1027672124_state @ P_41 )
        = bot_bo70021908tate_o )
    <=> ! [X_5: hoare_1167836817_state] :
          ~ ( P_41 @ X_5 ) ) )).

thf(fact_72_empty__iff,axiom,(
    ! [C_51: pname] :
      ~ ( member_pname @ C_51 @ bot_bot_pname_o ) )).

thf(fact_73_empty__iff,axiom,(
    ! [C_51: hoare_1167836817_state] :
      ~ ( member2058392318_state @ C_51 @ bot_bo70021908tate_o ) )).

thf(fact_74_empty__Collect__eq,axiom,(
    ! [P_40: pname > $o] :
      ( ( bot_bot_pname_o
        = ( collect_pname @ P_40 ) )
    <=> ! [X_5: pname] :
          ~ ( P_40 @ X_5 ) ) )).

thf(fact_75_empty__Collect__eq,axiom,(
    ! [P_40: ( pname > $o ) > $o] :
      ( ( bot_bot_pname_o_o
        = ( collect_pname_o @ P_40 ) )
    <=> ! [X_5: pname > $o] :
          ~ ( P_40 @ X_5 ) ) )).

thf(fact_76_empty__Collect__eq,axiom,(
    ! [P_40: ( hoare_1167836817_state > $o ) > $o] :
      ( ( bot_bo691907561te_o_o
        = ( collec269976083tate_o @ P_40 ) )
    <=> ! [X_5: hoare_1167836817_state > $o] :
          ~ ( P_40 @ X_5 ) ) )).

thf(fact_77_empty__Collect__eq,axiom,(
    ! [P_40: hoare_1167836817_state > $o] :
      ( ( bot_bo70021908tate_o
        = ( collec1027672124_state @ P_40 ) )
    <=> ! [X_5: hoare_1167836817_state] :
          ~ ( P_40 @ X_5 ) ) )).

thf(fact_78_ex__in__conv,axiom,(
    ! [A_184: pname > $o] :
      ( ? [X_5: pname] :
          ( member_pname @ X_5 @ A_184 )
    <=> ( A_184 != bot_bot_pname_o ) ) )).

thf(fact_79_ex__in__conv,axiom,(
    ! [A_184: hoare_1167836817_state > $o] :
      ( ? [X_5: hoare_1167836817_state] :
          ( member2058392318_state @ X_5 @ A_184 )
    <=> ( A_184 != bot_bo70021908tate_o ) ) )).

thf(fact_80_all__not__in__conv,axiom,(
    ! [A_183: pname > $o] :
      ( ! [X_5: pname] :
          ~ ( member_pname @ X_5 @ A_183 )
    <=> ( A_183 = bot_bot_pname_o ) ) )).

thf(fact_81_all__not__in__conv,axiom,(
    ! [A_183: hoare_1167836817_state > $o] :
      ( ! [X_5: hoare_1167836817_state] :
          ~ ( member2058392318_state @ X_5 @ A_183 )
    <=> ( A_183 = bot_bo70021908tate_o ) ) )).

thf(fact_82_empty__def,axiom,
    ( bot_bot_pname_o
    = ( collect_pname
      @ ^ [X_5: pname] : $false ) )).

thf(fact_83_empty__def,axiom,
    ( bot_bot_pname_o_o
    = ( collect_pname_o
      @ ^ [X_5: pname > $o] : $false ) )).

thf(fact_84_empty__def,axiom,
    ( bot_bo691907561te_o_o
    = ( collec269976083tate_o
      @ ^ [X_5: hoare_1167836817_state > $o] : $false ) )).

thf(fact_85_empty__def,axiom,
    ( bot_bo70021908tate_o
    = ( collec1027672124_state
      @ ^ [X_5: hoare_1167836817_state] : $false ) )).

thf(fact_86_insert__absorb,axiom,(
    ! [A_182: pname,A_181: pname > $o] :
      ( ( member_pname @ A_182 @ A_181 )
     => ( ( insert_pname @ A_182 @ A_181 )
        = A_181 ) ) )).

thf(fact_87_insert__absorb,axiom,(
    ! [A_182: hoare_1167836817_state,A_181: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ A_182 @ A_181 )
     => ( ( insert2134838167_state @ A_182 @ A_181 )
        = A_181 ) ) )).

thf(fact_88_insertI2,axiom,(
    ! [B_123: pname,A_180: pname,B_122: pname > $o] :
      ( ( member_pname @ A_180 @ B_122 )
     => ( member_pname @ A_180 @ ( insert_pname @ B_123 @ B_122 ) ) ) )).

thf(fact_89_insertI2,axiom,(
    ! [B_123: hoare_1167836817_state,A_180: hoare_1167836817_state,B_122: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ A_180 @ B_122 )
     => ( member2058392318_state @ A_180 @ ( insert2134838167_state @ B_123 @ B_122 ) ) ) )).

thf(fact_90_insert__ident,axiom,(
    ! [B_121: pname > $o,X_97: pname,A_179: pname > $o] :
      ( ~ ( member_pname @ X_97 @ A_179 )
     => ( ~ ( member_pname @ X_97 @ B_121 )
       => ( ( ( insert_pname @ X_97 @ A_179 )
            = ( insert_pname @ X_97 @ B_121 ) )
        <=> ( A_179 = B_121 ) ) ) ) )).

thf(fact_91_insert__ident,axiom,(
    ! [B_121: hoare_1167836817_state > $o,X_97: hoare_1167836817_state,A_179: hoare_1167836817_state > $o] :
      ( ~ ( member2058392318_state @ X_97 @ A_179 )
     => ( ~ ( member2058392318_state @ X_97 @ B_121 )
       => ( ( ( insert2134838167_state @ X_97 @ A_179 )
            = ( insert2134838167_state @ X_97 @ B_121 ) )
        <=> ( A_179 = B_121 ) ) ) ) )).

thf(fact_92_insert__code,axiom,(
    ! [Y_48: pname,A_178: pname > $o,X_96: pname] :
      ( ( insert_pname @ Y_48 @ A_178 @ X_96 )
    <=> ( ( Y_48 = X_96 )
        | ( A_178 @ X_96 ) ) ) )).

thf(fact_93_insert__code,axiom,(
    ! [Y_48: hoare_1167836817_state,A_178: hoare_1167836817_state > $o,X_96: hoare_1167836817_state] :
      ( ( insert2134838167_state @ Y_48 @ A_178 @ X_96 )
    <=> ( ( Y_48 = X_96 )
        | ( A_178 @ X_96 ) ) ) )).

thf(fact_94_insert__iff,axiom,(
    ! [A_177: pname,B_120: pname,A_176: pname > $o] :
      ( ( member_pname @ A_177 @ ( insert_pname @ B_120 @ A_176 ) )
    <=> ( ( A_177 = B_120 )
        | ( member_pname @ A_177 @ A_176 ) ) ) )).

thf(fact_95_insert__iff,axiom,(
    ! [A_177: hoare_1167836817_state,B_120: hoare_1167836817_state,A_176: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ A_177 @ ( insert2134838167_state @ B_120 @ A_176 ) )
    <=> ( ( A_177 = B_120 )
        | ( member2058392318_state @ A_177 @ A_176 ) ) ) )).

thf(fact_96_insert__commute,axiom,(
    ! [X_95: pname,Y_47: pname,A_175: pname > $o] :
      ( ( insert_pname @ X_95 @ ( insert_pname @ Y_47 @ A_175 ) )
      = ( insert_pname @ Y_47 @ ( insert_pname @ X_95 @ A_175 ) ) ) )).

thf(fact_97_insert__commute,axiom,(
    ! [X_95: hoare_1167836817_state,Y_47: hoare_1167836817_state,A_175: hoare_1167836817_state > $o] :
      ( ( insert2134838167_state @ X_95 @ ( insert2134838167_state @ Y_47 @ A_175 ) )
      = ( insert2134838167_state @ Y_47 @ ( insert2134838167_state @ X_95 @ A_175 ) ) ) )).

thf(fact_98_insert__absorb2,axiom,(
    ! [X_94: pname,A_174: pname > $o] :
      ( ( insert_pname @ X_94 @ ( insert_pname @ X_94 @ A_174 ) )
      = ( insert_pname @ X_94 @ A_174 ) ) )).

thf(fact_99_insert__absorb2,axiom,(
    ! [X_94: hoare_1167836817_state,A_174: hoare_1167836817_state > $o] :
      ( ( insert2134838167_state @ X_94 @ ( insert2134838167_state @ X_94 @ A_174 ) )
      = ( insert2134838167_state @ X_94 @ A_174 ) ) )).

thf(fact_100_insert__Collect,axiom,(
    ! [A_173: pname,P_39: pname > $o] :
      ( ( insert_pname @ A_173 @ ( collect_pname @ P_39 ) )
      = ( collect_pname
        @ ^ [U_2: pname] :
            ( (=>) @ ( (~) @ ( U_2 = A_173 ) ) @ ( P_39 @ U_2 ) ) ) ) )).

thf(fact_101_insert__Collect,axiom,(
    ! [A_173: pname > $o,P_39: ( pname > $o ) > $o] :
      ( ( insert_pname_o @ A_173 @ ( collect_pname_o @ P_39 ) )
      = ( collect_pname_o
        @ ^ [U_2: pname > $o] :
            ( (=>) @ ( (~) @ ( U_2 = A_173 ) ) @ ( P_39 @ U_2 ) ) ) ) )).

thf(fact_102_insert__Collect,axiom,(
    ! [A_173: hoare_1167836817_state > $o,P_39: ( hoare_1167836817_state > $o ) > $o] :
      ( ( insert999278200tate_o @ A_173 @ ( collec269976083tate_o @ P_39 ) )
      = ( collec269976083tate_o
        @ ^ [U_2: hoare_1167836817_state > $o] :
            ( (=>) @ ( (~) @ ( U_2 = A_173 ) ) @ ( P_39 @ U_2 ) ) ) ) )).

thf(fact_103_insert__Collect,axiom,(
    ! [A_173: hoare_1167836817_state,P_39: hoare_1167836817_state > $o] :
      ( ( insert2134838167_state @ A_173 @ ( collec1027672124_state @ P_39 ) )
      = ( collec1027672124_state
        @ ^ [U_2: hoare_1167836817_state] :
            ( (=>) @ ( (~) @ ( U_2 = A_173 ) ) @ ( P_39 @ U_2 ) ) ) ) )).

thf(fact_104_insert__compr,axiom,(
    ! [A_172: pname,B_119: pname > $o] :
      ( ( insert_pname @ A_172 @ B_119 )
      = ( collect_pname
        @ ^ [X_5: pname] :
            ( (|) @ ( X_5 = A_172 ) @ ( member_pname @ X_5 @ B_119 ) ) ) ) )).

thf(fact_105_insert__compr,axiom,(
    ! [A_172: pname > $o,B_119: ( pname > $o ) > $o] :
      ( ( insert_pname_o @ A_172 @ B_119 )
      = ( collect_pname_o
        @ ^ [X_5: pname > $o] :
            ( (|) @ ( X_5 = A_172 ) @ ( member_pname_o @ X_5 @ B_119 ) ) ) ) )).

thf(fact_106_insert__compr,axiom,(
    ! [A_172: hoare_1167836817_state > $o,B_119: ( hoare_1167836817_state > $o ) > $o] :
      ( ( insert999278200tate_o @ A_172 @ B_119 )
      = ( collec269976083tate_o
        @ ^ [X_5: hoare_1167836817_state > $o] :
            ( (|) @ ( X_5 = A_172 ) @ ( member864234961tate_o @ X_5 @ B_119 ) ) ) ) )).

thf(fact_107_insert__compr,axiom,(
    ! [A_172: hoare_1167836817_state,B_119: hoare_1167836817_state > $o] :
      ( ( insert2134838167_state @ A_172 @ B_119 )
      = ( collec1027672124_state
        @ ^ [X_5: hoare_1167836817_state] :
            ( (|) @ ( X_5 = A_172 ) @ ( member2058392318_state @ X_5 @ B_119 ) ) ) ) )).

thf(fact_108_insertI1,axiom,(
    ! [A_171: pname,B_118: pname > $o] :
      ( member_pname @ A_171 @ ( insert_pname @ A_171 @ B_118 ) ) )).

thf(fact_109_insertI1,axiom,(
    ! [A_171: hoare_1167836817_state,B_118: hoare_1167836817_state > $o] :
      ( member2058392318_state @ A_171 @ ( insert2134838167_state @ A_171 @ B_118 ) ) )).

thf(fact_110_equalityE,axiom,(
    ! [A_170: pname > $o,B_117: pname > $o] :
      ( ( A_170 = B_117 )
     => ~ ( ( ord_less_eq_pname_o @ A_170 @ B_117 )
         => ~ ( ord_less_eq_pname_o @ B_117 @ A_170 ) ) ) )).

thf(fact_111_equalityE,axiom,(
    ! [A_170: hoare_1167836817_state > $o,B_117: hoare_1167836817_state > $o] :
      ( ( A_170 = B_117 )
     => ~ ( ( ord_le827224136tate_o @ A_170 @ B_117 )
         => ~ ( ord_le827224136tate_o @ B_117 @ A_170 ) ) ) )).

thf(fact_112_subset__trans,axiom,(
    ! [C_50: pname > $o,A_169: pname > $o,B_116: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_169 @ B_116 )
     => ( ( ord_less_eq_pname_o @ B_116 @ C_50 )
       => ( ord_less_eq_pname_o @ A_169 @ C_50 ) ) ) )).

thf(fact_113_subset__trans,axiom,(
    ! [C_50: hoare_1167836817_state > $o,A_169: hoare_1167836817_state > $o,B_116: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_169 @ B_116 )
     => ( ( ord_le827224136tate_o @ B_116 @ C_50 )
       => ( ord_le827224136tate_o @ A_169 @ C_50 ) ) ) )).

thf(fact_114_set__mp,axiom,(
    ! [X_93: pname,A_168: pname > $o,B_115: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_168 @ B_115 )
     => ( ( member_pname @ X_93 @ A_168 )
       => ( member_pname @ X_93 @ B_115 ) ) ) )).

thf(fact_115_set__mp,axiom,(
    ! [X_93: hoare_1167836817_state,A_168: hoare_1167836817_state > $o,B_115: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_168 @ B_115 )
     => ( ( member2058392318_state @ X_93 @ A_168 )
       => ( member2058392318_state @ X_93 @ B_115 ) ) ) )).

thf(fact_116_set__rev__mp,axiom,(
    ! [B_114: pname > $o,X_92: pname,A_167: pname > $o] :
      ( ( member_pname @ X_92 @ A_167 )
     => ( ( ord_less_eq_pname_o @ A_167 @ B_114 )
       => ( member_pname @ X_92 @ B_114 ) ) ) )).

thf(fact_117_set__rev__mp,axiom,(
    ! [B_114: hoare_1167836817_state > $o,X_92: hoare_1167836817_state,A_167: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ X_92 @ A_167 )
     => ( ( ord_le827224136tate_o @ A_167 @ B_114 )
       => ( member2058392318_state @ X_92 @ B_114 ) ) ) )).

thf(fact_118_in__mono,axiom,(
    ! [X_91: pname,A_166: pname > $o,B_113: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_166 @ B_113 )
     => ( ( member_pname @ X_91 @ A_166 )
       => ( member_pname @ X_91 @ B_113 ) ) ) )).

thf(fact_119_in__mono,axiom,(
    ! [X_91: hoare_1167836817_state,A_166: hoare_1167836817_state > $o,B_113: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_166 @ B_113 )
     => ( ( member2058392318_state @ X_91 @ A_166 )
       => ( member2058392318_state @ X_91 @ B_113 ) ) ) )).

thf(fact_120_equalityD2,axiom,(
    ! [A_165: pname > $o,B_112: pname > $o] :
      ( ( A_165 = B_112 )
     => ( ord_less_eq_pname_o @ B_112 @ A_165 ) ) )).

thf(fact_121_equalityD2,axiom,(
    ! [A_165: hoare_1167836817_state > $o,B_112: hoare_1167836817_state > $o] :
      ( ( A_165 = B_112 )
     => ( ord_le827224136tate_o @ B_112 @ A_165 ) ) )).

thf(fact_122_equalityD1,axiom,(
    ! [A_164: pname > $o,B_111: pname > $o] :
      ( ( A_164 = B_111 )
     => ( ord_less_eq_pname_o @ A_164 @ B_111 ) ) )).

thf(fact_123_equalityD1,axiom,(
    ! [A_164: hoare_1167836817_state > $o,B_111: hoare_1167836817_state > $o] :
      ( ( A_164 = B_111 )
     => ( ord_le827224136tate_o @ A_164 @ B_111 ) ) )).

thf(fact_124_set__eq__subset,axiom,(
    ! [A_163: pname > $o,B_110: pname > $o] :
      ( ( A_163 = B_110 )
    <=> ( ( ord_less_eq_pname_o @ A_163 @ B_110 )
        & ( ord_less_eq_pname_o @ B_110 @ A_163 ) ) ) )).

thf(fact_125_set__eq__subset,axiom,(
    ! [A_163: hoare_1167836817_state > $o,B_110: hoare_1167836817_state > $o] :
      ( ( A_163 = B_110 )
    <=> ( ( ord_le827224136tate_o @ A_163 @ B_110 )
        & ( ord_le827224136tate_o @ B_110 @ A_163 ) ) ) )).

thf(fact_126_subset__refl,axiom,(
    ! [A_162: pname > $o] :
      ( ord_less_eq_pname_o @ A_162 @ A_162 ) )).

thf(fact_127_subset__refl,axiom,(
    ! [A_162: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ A_162 @ A_162 ) )).

thf(fact_128_rev__image__eqI,axiom,(
    ! [B_109: pname,F_57: hoare_1167836817_state > pname,X_90: hoare_1167836817_state,A_161: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ X_90 @ A_161 )
     => ( ( B_109
          = ( F_57 @ X_90 ) )
       => ( member_pname @ B_109 @ ( image_8178176_pname @ F_57 @ A_161 ) ) ) ) )).

thf(fact_129_rev__image__eqI,axiom,(
    ! [B_109: hoare_1167836817_state,F_57: pname > hoare_1167836817_state,X_90: pname,A_161: pname > $o] :
      ( ( member_pname @ X_90 @ A_161 )
     => ( ( B_109
          = ( F_57 @ X_90 ) )
       => ( member2058392318_state @ B_109 @ ( image_575578384_state @ F_57 @ A_161 ) ) ) ) )).

thf(fact_130_imageI,axiom,(
    ! [F_56: hoare_1167836817_state > pname,X_89: hoare_1167836817_state,A_160: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ X_89 @ A_160 )
     => ( member_pname @ ( F_56 @ X_89 ) @ ( image_8178176_pname @ F_56 @ A_160 ) ) ) )).

thf(fact_131_imageI,axiom,(
    ! [F_56: pname > hoare_1167836817_state,X_89: pname,A_160: pname > $o] :
      ( ( member_pname @ X_89 @ A_160 )
     => ( member2058392318_state @ ( F_56 @ X_89 ) @ ( image_575578384_state @ F_56 @ A_160 ) ) ) )).

thf(fact_132_image__iff,axiom,(
    ! [Z_21: hoare_1167836817_state,F_55: pname > hoare_1167836817_state,A_159: pname > $o] :
      ( ( member2058392318_state @ Z_21 @ ( image_575578384_state @ F_55 @ A_159 ) )
    <=> ? [X_5: pname] :
          ( ( member_pname @ X_5 @ A_159 )
          & ( Z_21
            = ( F_55 @ X_5 ) ) ) ) )).

thf(fact_133_finite__Collect__disjI,axiom,(
    ! [P_38: ( pname > $o ) > $o,Q_23: ( pname > $o ) > $o] :
      ( ( finite297249702name_o
        @ ( collect_pname_o
          @ ^ [X_5: pname > $o] :
              ( (|) @ ( P_38 @ X_5 ) @ ( Q_23 @ X_5 ) ) ) )
    <=> ( ( finite297249702name_o @ ( collect_pname_o @ P_38 ) )
        & ( finite297249702name_o @ ( collect_pname_o @ Q_23 ) ) ) ) )).

thf(fact_134_finite__Collect__disjI,axiom,(
    ! [P_38: ( hoare_1167836817_state > $o ) > $o,Q_23: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o
        @ ( collec269976083tate_o
          @ ^ [X_5: hoare_1167836817_state > $o] :
              ( (|) @ ( P_38 @ X_5 ) @ ( Q_23 @ X_5 ) ) ) )
    <=> ( ( finite1380128977tate_o @ ( collec269976083tate_o @ P_38 ) )
        & ( finite1380128977tate_o @ ( collec269976083tate_o @ Q_23 ) ) ) ) )).

thf(fact_135_finite__Collect__disjI,axiom,(
    ! [P_38: hoare_1167836817_state > $o,Q_23: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state
        @ ( collec1027672124_state
          @ ^ [X_5: hoare_1167836817_state] :
              ( (|) @ ( P_38 @ X_5 ) @ ( Q_23 @ X_5 ) ) ) )
    <=> ( ( finite1084549118_state @ ( collec1027672124_state @ P_38 ) )
        & ( finite1084549118_state @ ( collec1027672124_state @ Q_23 ) ) ) ) )).

thf(fact_136_finite__Collect__disjI,axiom,(
    ! [P_38: pname > $o,Q_23: pname > $o] :
      ( ( finite_finite_pname
        @ ( collect_pname
          @ ^ [X_5: pname] :
              ( (|) @ ( P_38 @ X_5 ) @ ( Q_23 @ X_5 ) ) ) )
    <=> ( ( finite_finite_pname @ ( collect_pname @ P_38 ) )
        & ( finite_finite_pname @ ( collect_pname @ Q_23 ) ) ) ) )).

thf(fact_137_insert__compr__raw,axiom,(
    ! [X_5: pname,Xa: pname > $o] :
      ( ( insert_pname @ X_5 @ Xa )
      = ( collect_pname
        @ ^ [Y_2: pname] :
            ( (|) @ ( Y_2 = X_5 ) @ ( member_pname @ Y_2 @ Xa ) ) ) ) )).

thf(fact_138_insert__compr__raw,axiom,(
    ! [X_5: pname > $o,Xa: ( pname > $o ) > $o] :
      ( ( insert_pname_o @ X_5 @ Xa )
      = ( collect_pname_o
        @ ^ [Y_2: pname > $o] :
            ( (|) @ ( Y_2 = X_5 ) @ ( member_pname_o @ Y_2 @ Xa ) ) ) ) )).

thf(fact_139_insert__compr__raw,axiom,(
    ! [X_5: hoare_1167836817_state > $o,Xa: ( hoare_1167836817_state > $o ) > $o] :
      ( ( insert999278200tate_o @ X_5 @ Xa )
      = ( collec269976083tate_o
        @ ^ [Y_2: hoare_1167836817_state > $o] :
            ( (|) @ ( Y_2 = X_5 ) @ ( member864234961tate_o @ Y_2 @ Xa ) ) ) ) )).

thf(fact_140_insert__compr__raw,axiom,(
    ! [X_5: hoare_1167836817_state,Xa: hoare_1167836817_state > $o] :
      ( ( insert2134838167_state @ X_5 @ Xa )
      = ( collec1027672124_state
        @ ^ [Y_2: hoare_1167836817_state] :
            ( (|) @ ( Y_2 = X_5 ) @ ( member2058392318_state @ Y_2 @ Xa ) ) ) ) )).

thf(fact_141_singleton__inject,axiom,(
    ! [A_158: pname,B_108: pname] :
      ( ( ( insert_pname @ A_158 @ bot_bot_pname_o )
        = ( insert_pname @ B_108 @ bot_bot_pname_o ) )
     => ( A_158 = B_108 ) ) )).

thf(fact_142_singleton__inject,axiom,(
    ! [A_158: hoare_1167836817_state,B_108: hoare_1167836817_state] :
      ( ( ( insert2134838167_state @ A_158 @ bot_bo70021908tate_o )
        = ( insert2134838167_state @ B_108 @ bot_bo70021908tate_o ) )
     => ( A_158 = B_108 ) ) )).

thf(fact_143_singletonE,axiom,(
    ! [B_107: pname,A_157: pname] :
      ( ( member_pname @ B_107 @ ( insert_pname @ A_157 @ bot_bot_pname_o ) )
     => ( B_107 = A_157 ) ) )).

thf(fact_144_singletonE,axiom,(
    ! [B_107: hoare_1167836817_state,A_157: hoare_1167836817_state] :
      ( ( member2058392318_state @ B_107 @ ( insert2134838167_state @ A_157 @ bot_bo70021908tate_o ) )
     => ( B_107 = A_157 ) ) )).

thf(fact_145_doubleton__eq__iff,axiom,(
    ! [A_156: pname,B_106: pname,C_49: pname,D_6: pname] :
      ( ( ( insert_pname @ A_156 @ ( insert_pname @ B_106 @ bot_bot_pname_o ) )
        = ( insert_pname @ C_49 @ ( insert_pname @ D_6 @ bot_bot_pname_o ) ) )
    <=> ( ( ( A_156 = C_49 )
          & ( B_106 = D_6 ) )
        | ( ( A_156 = D_6 )
          & ( B_106 = C_49 ) ) ) ) )).

thf(fact_146_doubleton__eq__iff,axiom,(
    ! [A_156: hoare_1167836817_state,B_106: hoare_1167836817_state,C_49: hoare_1167836817_state,D_6: hoare_1167836817_state] :
      ( ( ( insert2134838167_state @ A_156 @ ( insert2134838167_state @ B_106 @ bot_bo70021908tate_o ) )
        = ( insert2134838167_state @ C_49 @ ( insert2134838167_state @ D_6 @ bot_bo70021908tate_o ) ) )
    <=> ( ( ( A_156 = C_49 )
          & ( B_106 = D_6 ) )
        | ( ( A_156 = D_6 )
          & ( B_106 = C_49 ) ) ) ) )).

thf(fact_147_singleton__iff,axiom,(
    ! [B_105: pname,A_155: pname] :
      ( ( member_pname @ B_105 @ ( insert_pname @ A_155 @ bot_bot_pname_o ) )
    <=> ( B_105 = A_155 ) ) )).

thf(fact_148_singleton__iff,axiom,(
    ! [B_105: hoare_1167836817_state,A_155: hoare_1167836817_state] :
      ( ( member2058392318_state @ B_105 @ ( insert2134838167_state @ A_155 @ bot_bo70021908tate_o ) )
    <=> ( B_105 = A_155 ) ) )).

thf(fact_149_insert__not__empty,axiom,(
    ! [A_154: pname,A_153: pname > $o] :
      ( ( insert_pname @ A_154 @ A_153 )
     != bot_bot_pname_o ) )).

thf(fact_150_insert__not__empty,axiom,(
    ! [A_154: hoare_1167836817_state,A_153: hoare_1167836817_state > $o] :
      ( ( insert2134838167_state @ A_154 @ A_153 )
     != bot_bo70021908tate_o ) )).

thf(fact_151_empty__not__insert,axiom,(
    ! [A_152: pname,A_151: pname > $o] :
      ( bot_bot_pname_o
     != ( insert_pname @ A_152 @ A_151 ) ) )).

thf(fact_152_empty__not__insert,axiom,(
    ! [A_152: hoare_1167836817_state,A_151: hoare_1167836817_state > $o] :
      ( bot_bo70021908tate_o
     != ( insert2134838167_state @ A_152 @ A_151 ) ) )).

thf(fact_153_finite__insert,axiom,(
    ! [A_150: pname > $o,A_149: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ ( insert_pname_o @ A_150 @ A_149 ) )
    <=> ( finite297249702name_o @ A_149 ) ) )).

thf(fact_154_finite__insert,axiom,(
    ! [A_150: hoare_1167836817_state > $o,A_149: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ ( insert999278200tate_o @ A_150 @ A_149 ) )
    <=> ( finite1380128977tate_o @ A_149 ) ) )).

thf(fact_155_finite__insert,axiom,(
    ! [A_150: pname,A_149: pname > $o] :
      ( ( finite_finite_pname @ ( insert_pname @ A_150 @ A_149 ) )
    <=> ( finite_finite_pname @ A_149 ) ) )).

thf(fact_156_finite__insert,axiom,(
    ! [A_150: hoare_1167836817_state,A_149: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ ( insert2134838167_state @ A_150 @ A_149 ) )
    <=> ( finite1084549118_state @ A_149 ) ) )).

thf(fact_157_subset__empty,axiom,(
    ! [A_148: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_148 @ bot_bot_pname_o )
    <=> ( A_148 = bot_bot_pname_o ) ) )).

thf(fact_158_subset__empty,axiom,(
    ! [A_148: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_148 @ bot_bo70021908tate_o )
    <=> ( A_148 = bot_bo70021908tate_o ) ) )).

thf(fact_159_image__is__empty,axiom,(
    ! [F_54: hoare_1167836817_state > pname,A_147: hoare_1167836817_state > $o] :
      ( ( ( image_8178176_pname @ F_54 @ A_147 )
        = bot_bot_pname_o )
    <=> ( A_147 = bot_bo70021908tate_o ) ) )).

thf(fact_160_image__is__empty,axiom,(
    ! [F_54: pname > hoare_1167836817_state,A_147: pname > $o] :
      ( ( ( image_575578384_state @ F_54 @ A_147 )
        = bot_bo70021908tate_o )
    <=> ( A_147 = bot_bot_pname_o ) ) )).

thf(fact_161_image__empty,axiom,(
    ! [F_53: hoare_1167836817_state > pname] :
      ( ( image_8178176_pname @ F_53 @ bot_bo70021908tate_o )
      = bot_bot_pname_o ) )).

thf(fact_162_image__empty,axiom,(
    ! [F_53: pname > hoare_1167836817_state] :
      ( ( image_575578384_state @ F_53 @ bot_bot_pname_o )
      = bot_bo70021908tate_o ) )).

thf(fact_163_empty__is__image,axiom,(
    ! [F_52: hoare_1167836817_state > pname,A_146: hoare_1167836817_state > $o] :
      ( ( bot_bot_pname_o
        = ( image_8178176_pname @ F_52 @ A_146 ) )
    <=> ( A_146 = bot_bo70021908tate_o ) ) )).

thf(fact_164_empty__is__image,axiom,(
    ! [F_52: pname > hoare_1167836817_state,A_146: pname > $o] :
      ( ( bot_bo70021908tate_o
        = ( image_575578384_state @ F_52 @ A_146 ) )
    <=> ( A_146 = bot_bot_pname_o ) ) )).

thf(fact_165_finite__subset,axiom,(
    ! [A_145: ( pname > $o ) > $o,B_104: ( pname > $o ) > $o] :
      ( ( ord_le1205211808me_o_o @ A_145 @ B_104 )
     => ( ( finite297249702name_o @ B_104 )
       => ( finite297249702name_o @ A_145 ) ) ) )).

thf(fact_166_finite__subset,axiom,(
    ! [A_145: ( hoare_1167836817_state > $o ) > $o,B_104: ( hoare_1167836817_state > $o ) > $o] :
      ( ( ord_le741939125te_o_o @ A_145 @ B_104 )
     => ( ( finite1380128977tate_o @ B_104 )
       => ( finite1380128977tate_o @ A_145 ) ) ) )).

thf(fact_167_finite__subset,axiom,(
    ! [A_145: hoare_1167836817_state > $o,B_104: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_145 @ B_104 )
     => ( ( finite1084549118_state @ B_104 )
       => ( finite1084549118_state @ A_145 ) ) ) )).

thf(fact_168_finite__subset,axiom,(
    ! [A_145: pname > $o,B_104: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_145 @ B_104 )
     => ( ( finite_finite_pname @ B_104 )
       => ( finite_finite_pname @ A_145 ) ) ) )).

thf(fact_169_rev__finite__subset,axiom,(
    ! [A_144: ( pname > $o ) > $o,B_103: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ B_103 )
     => ( ( ord_le1205211808me_o_o @ A_144 @ B_103 )
       => ( finite297249702name_o @ A_144 ) ) ) )).

thf(fact_170_rev__finite__subset,axiom,(
    ! [A_144: ( hoare_1167836817_state > $o ) > $o,B_103: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ B_103 )
     => ( ( ord_le741939125te_o_o @ A_144 @ B_103 )
       => ( finite1380128977tate_o @ A_144 ) ) ) )).

thf(fact_171_rev__finite__subset,axiom,(
    ! [A_144: hoare_1167836817_state > $o,B_103: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ B_103 )
     => ( ( ord_le827224136tate_o @ A_144 @ B_103 )
       => ( finite1084549118_state @ A_144 ) ) ) )).

thf(fact_172_rev__finite__subset,axiom,(
    ! [A_144: pname > $o,B_103: pname > $o] :
      ( ( finite_finite_pname @ B_103 )
     => ( ( ord_less_eq_pname_o @ A_144 @ B_103 )
       => ( finite_finite_pname @ A_144 ) ) ) )).

thf(fact_173_insert__mono,axiom,(
    ! [A_143: pname,C_48: pname > $o,D_5: pname > $o] :
      ( ( ord_less_eq_pname_o @ C_48 @ D_5 )
     => ( ord_less_eq_pname_o @ ( insert_pname @ A_143 @ C_48 ) @ ( insert_pname @ A_143 @ D_5 ) ) ) )).

thf(fact_174_insert__mono,axiom,(
    ! [A_143: hoare_1167836817_state,C_48: hoare_1167836817_state > $o,D_5: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ C_48 @ D_5 )
     => ( ord_le827224136tate_o @ ( insert2134838167_state @ A_143 @ C_48 ) @ ( insert2134838167_state @ A_143 @ D_5 ) ) ) )).

thf(fact_175_mem__def,axiom,(
    ! [X_88: pname,A_142: pname > $o] :
      ( ( member_pname @ X_88 @ A_142 )
    <=> ( A_142 @ X_88 ) ) )).

thf(fact_176_mem__def,axiom,(
    ! [X_88: hoare_1167836817_state,A_142: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ X_88 @ A_142 )
    <=> ( A_142 @ X_88 ) ) )).

thf(fact_177_Collect__def,axiom,(
    ! [P_37: hoare_1167836817_state > $o] :
      ( ( collec1027672124_state @ P_37 )
      = P_37 ) )).

thf(fact_178_Collect__def,axiom,(
    ! [P_37: pname > $o] :
      ( ( collect_pname @ P_37 )
      = P_37 ) )).

thf(fact_179_Collect__def,axiom,(
    ! [P_37: ( pname > $o ) > $o] :
      ( ( collect_pname_o @ P_37 )
      = P_37 ) )).

thf(fact_180_Collect__def,axiom,(
    ! [P_37: ( hoare_1167836817_state > $o ) > $o] :
      ( ( collec269976083tate_o @ P_37 )
      = P_37 ) )).

thf(fact_181_subset__insertI2,axiom,(
    ! [B_102: pname,A_141: pname > $o,B_101: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_141 @ B_101 )
     => ( ord_less_eq_pname_o @ A_141 @ ( insert_pname @ B_102 @ B_101 ) ) ) )).

thf(fact_182_subset__insertI2,axiom,(
    ! [B_102: hoare_1167836817_state,A_141: hoare_1167836817_state > $o,B_101: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_141 @ B_101 )
     => ( ord_le827224136tate_o @ A_141 @ ( insert2134838167_state @ B_102 @ B_101 ) ) ) )).

thf(fact_183_subset__insert,axiom,(
    ! [B_100: pname > $o,X_87: pname,A_140: pname > $o] :
      ( ~ ( member_pname @ X_87 @ A_140 )
     => ( ( ord_less_eq_pname_o @ A_140 @ ( insert_pname @ X_87 @ B_100 ) )
      <=> ( ord_less_eq_pname_o @ A_140 @ B_100 ) ) ) )).

thf(fact_184_subset__insert,axiom,(
    ! [B_100: hoare_1167836817_state > $o,X_87: hoare_1167836817_state,A_140: hoare_1167836817_state > $o] :
      ( ~ ( member2058392318_state @ X_87 @ A_140 )
     => ( ( ord_le827224136tate_o @ A_140 @ ( insert2134838167_state @ X_87 @ B_100 ) )
      <=> ( ord_le827224136tate_o @ A_140 @ B_100 ) ) ) )).

thf(fact_185_insert__subset,axiom,(
    ! [X_86: pname,A_139: pname > $o,B_99: pname > $o] :
      ( ( ord_less_eq_pname_o @ ( insert_pname @ X_86 @ A_139 ) @ B_99 )
    <=> ( ( member_pname @ X_86 @ B_99 )
        & ( ord_less_eq_pname_o @ A_139 @ B_99 ) ) ) )).

thf(fact_186_insert__subset,axiom,(
    ! [X_86: hoare_1167836817_state,A_139: hoare_1167836817_state > $o,B_99: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ ( insert2134838167_state @ X_86 @ A_139 ) @ B_99 )
    <=> ( ( member2058392318_state @ X_86 @ B_99 )
        & ( ord_le827224136tate_o @ A_139 @ B_99 ) ) ) )).

thf(fact_187_subset__insertI,axiom,(
    ! [B_98: pname > $o,A_138: pname] :
      ( ord_less_eq_pname_o @ B_98 @ ( insert_pname @ A_138 @ B_98 ) ) )).

thf(fact_188_subset__insertI,axiom,(
    ! [B_98: hoare_1167836817_state > $o,A_138: hoare_1167836817_state] :
      ( ord_le827224136tate_o @ B_98 @ ( insert2134838167_state @ A_138 @ B_98 ) ) )).

thf(fact_189_insert__image,axiom,(
    ! [F_51: hoare_1167836817_state > pname,X_85: hoare_1167836817_state,A_137: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ X_85 @ A_137 )
     => ( ( insert_pname @ ( F_51 @ X_85 ) @ ( image_8178176_pname @ F_51 @ A_137 ) )
        = ( image_8178176_pname @ F_51 @ A_137 ) ) ) )).

thf(fact_190_insert__image,axiom,(
    ! [F_51: pname > hoare_1167836817_state,X_85: pname,A_137: pname > $o] :
      ( ( member_pname @ X_85 @ A_137 )
     => ( ( insert2134838167_state @ ( F_51 @ X_85 ) @ ( image_575578384_state @ F_51 @ A_137 ) )
        = ( image_575578384_state @ F_51 @ A_137 ) ) ) )).

thf(fact_191_image__insert,axiom,(
    ! [F_50: hoare_1167836817_state > pname,A_136: hoare_1167836817_state,B_97: hoare_1167836817_state > $o] :
      ( ( image_8178176_pname @ F_50 @ ( insert2134838167_state @ A_136 @ B_97 ) )
      = ( insert_pname @ ( F_50 @ A_136 ) @ ( image_8178176_pname @ F_50 @ B_97 ) ) ) )).

thf(fact_192_image__insert,axiom,(
    ! [F_50: pname > hoare_1167836817_state,A_136: pname,B_97: pname > $o] :
      ( ( image_575578384_state @ F_50 @ ( insert_pname @ A_136 @ B_97 ) )
      = ( insert2134838167_state @ ( F_50 @ A_136 ) @ ( image_575578384_state @ F_50 @ B_97 ) ) ) )).

thf(fact_193_image__mono,axiom,(
    ! [F_49: hoare_1167836817_state > pname,A_135: hoare_1167836817_state > $o,B_96: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_135 @ B_96 )
     => ( ord_less_eq_pname_o @ ( image_8178176_pname @ F_49 @ A_135 ) @ ( image_8178176_pname @ F_49 @ B_96 ) ) ) )).

thf(fact_194_image__mono,axiom,(
    ! [F_49: pname > hoare_1167836817_state,A_135: pname > $o,B_96: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_135 @ B_96 )
     => ( ord_le827224136tate_o @ ( image_575578384_state @ F_49 @ A_135 ) @ ( image_575578384_state @ F_49 @ B_96 ) ) ) )).

thf(fact_195_subset__image__iff,axiom,(
    ! [B_95: pname > $o,F_48: hoare_1167836817_state > pname,A_134: hoare_1167836817_state > $o] :
      ( ( ord_less_eq_pname_o @ B_95 @ ( image_8178176_pname @ F_48 @ A_134 ) )
    <=> ? [AA: hoare_1167836817_state > $o] :
          ( ( ord_le827224136tate_o @ AA @ A_134 )
          & ( B_95
            = ( image_8178176_pname @ F_48 @ AA ) ) ) ) )).

thf(fact_196_subset__image__iff,axiom,(
    ! [B_95: hoare_1167836817_state > $o,F_48: pname > hoare_1167836817_state,A_134: pname > $o] :
      ( ( ord_le827224136tate_o @ B_95 @ ( image_575578384_state @ F_48 @ A_134 ) )
    <=> ? [AA: pname > $o] :
          ( ( ord_less_eq_pname_o @ AA @ A_134 )
          & ( B_95
            = ( image_575578384_state @ F_48 @ AA ) ) ) ) )).

thf(fact_197_domI,axiom,(
    ! [M_2: pname > option_com,A_133: pname,B_94: com] :
      ( ( ( M_2 @ A_133 )
        = ( some_com @ B_94 ) )
     => ( member_pname @ A_133 @ ( dom_pname_com @ M_2 ) ) ) )).

thf(fact_198_Collect__conv__if,axiom,(
    ! [P_36: pname > $o,A_132: pname] :
      ( ( ( P_36 @ A_132 )
       => ( ( collect_pname
            @ ^ [X_5: pname] :
                ( (&) @ ( X_5 = A_132 ) @ ( P_36 @ X_5 ) ) )
          = ( insert_pname @ A_132 @ bot_bot_pname_o ) ) )
      & ( ~ ( P_36 @ A_132 )
       => ( ( collect_pname
            @ ^ [X_5: pname] :
                ( (&) @ ( X_5 = A_132 ) @ ( P_36 @ X_5 ) ) )
          = bot_bot_pname_o ) ) ) )).

thf(fact_199_Collect__conv__if,axiom,(
    ! [P_36: ( pname > $o ) > $o,A_132: pname > $o] :
      ( ( ( P_36 @ A_132 )
       => ( ( collect_pname_o
            @ ^ [X_5: pname > $o] :
                ( (&) @ ( X_5 = A_132 ) @ ( P_36 @ X_5 ) ) )
          = ( insert_pname_o @ A_132 @ bot_bot_pname_o_o ) ) )
      & ( ~ ( P_36 @ A_132 )
       => ( ( collect_pname_o
            @ ^ [X_5: pname > $o] :
                ( (&) @ ( X_5 = A_132 ) @ ( P_36 @ X_5 ) ) )
          = bot_bot_pname_o_o ) ) ) )).

thf(fact_200_Collect__conv__if,axiom,(
    ! [P_36: ( hoare_1167836817_state > $o ) > $o,A_132: hoare_1167836817_state > $o] :
      ( ( ( P_36 @ A_132 )
       => ( ( collec269976083tate_o
            @ ^ [X_5: hoare_1167836817_state > $o] :
                ( (&) @ ( X_5 = A_132 ) @ ( P_36 @ X_5 ) ) )
          = ( insert999278200tate_o @ A_132 @ bot_bo691907561te_o_o ) ) )
      & ( ~ ( P_36 @ A_132 )
       => ( ( collec269976083tate_o
            @ ^ [X_5: hoare_1167836817_state > $o] :
                ( (&) @ ( X_5 = A_132 ) @ ( P_36 @ X_5 ) ) )
          = bot_bo691907561te_o_o ) ) ) )).

thf(fact_201_Collect__conv__if,axiom,(
    ! [P_36: hoare_1167836817_state > $o,A_132: hoare_1167836817_state] :
      ( ( ( P_36 @ A_132 )
       => ( ( collec1027672124_state
            @ ^ [X_5: hoare_1167836817_state] :
                ( (&) @ ( X_5 = A_132 ) @ ( P_36 @ X_5 ) ) )
          = ( insert2134838167_state @ A_132 @ bot_bo70021908tate_o ) ) )
      & ( ~ ( P_36 @ A_132 )
       => ( ( collec1027672124_state
            @ ^ [X_5: hoare_1167836817_state] :
                ( (&) @ ( X_5 = A_132 ) @ ( P_36 @ X_5 ) ) )
          = bot_bo70021908tate_o ) ) ) )).

thf(fact_202_Collect__conv__if2,axiom,(
    ! [P_35: pname > $o,A_131: pname] :
      ( ( ( P_35 @ A_131 )
       => ( ( collect_pname
            @ ^ [X_5: pname] :
                ( (&) @ ( A_131 = X_5 ) @ ( P_35 @ X_5 ) ) )
          = ( insert_pname @ A_131 @ bot_bot_pname_o ) ) )
      & ( ~ ( P_35 @ A_131 )
       => ( ( collect_pname
            @ ^ [X_5: pname] :
                ( (&) @ ( A_131 = X_5 ) @ ( P_35 @ X_5 ) ) )
          = bot_bot_pname_o ) ) ) )).

thf(fact_203_Collect__conv__if2,axiom,(
    ! [P_35: ( pname > $o ) > $o,A_131: pname > $o] :
      ( ( ( P_35 @ A_131 )
       => ( ( collect_pname_o
            @ ^ [X_5: pname > $o] :
                ( (&) @ ( A_131 = X_5 ) @ ( P_35 @ X_5 ) ) )
          = ( insert_pname_o @ A_131 @ bot_bot_pname_o_o ) ) )
      & ( ~ ( P_35 @ A_131 )
       => ( ( collect_pname_o
            @ ^ [X_5: pname > $o] :
                ( (&) @ ( A_131 = X_5 ) @ ( P_35 @ X_5 ) ) )
          = bot_bot_pname_o_o ) ) ) )).

thf(fact_204_Collect__conv__if2,axiom,(
    ! [P_35: ( hoare_1167836817_state > $o ) > $o,A_131: hoare_1167836817_state > $o] :
      ( ( ( P_35 @ A_131 )
       => ( ( collec269976083tate_o
            @ ^ [X_5: hoare_1167836817_state > $o] :
                ( (&) @ ( A_131 = X_5 ) @ ( P_35 @ X_5 ) ) )
          = ( insert999278200tate_o @ A_131 @ bot_bo691907561te_o_o ) ) )
      & ( ~ ( P_35 @ A_131 )
       => ( ( collec269976083tate_o
            @ ^ [X_5: hoare_1167836817_state > $o] :
                ( (&) @ ( A_131 = X_5 ) @ ( P_35 @ X_5 ) ) )
          = bot_bo691907561te_o_o ) ) ) )).

thf(fact_205_Collect__conv__if2,axiom,(
    ! [P_35: hoare_1167836817_state > $o,A_131: hoare_1167836817_state] :
      ( ( ( P_35 @ A_131 )
       => ( ( collec1027672124_state
            @ ^ [X_5: hoare_1167836817_state] :
                ( (&) @ ( A_131 = X_5 ) @ ( P_35 @ X_5 ) ) )
          = ( insert2134838167_state @ A_131 @ bot_bo70021908tate_o ) ) )
      & ( ~ ( P_35 @ A_131 )
       => ( ( collec1027672124_state
            @ ^ [X_5: hoare_1167836817_state] :
                ( (&) @ ( A_131 = X_5 ) @ ( P_35 @ X_5 ) ) )
          = bot_bo70021908tate_o ) ) ) )).

thf(fact_206_singleton__conv,axiom,(
    ! [A_130: pname] :
      ( ( collect_pname
        @ ^ [X_5: pname] : ( X_5 = A_130 ) )
      = ( insert_pname @ A_130 @ bot_bot_pname_o ) ) )).

thf(fact_207_singleton__conv,axiom,(
    ! [A_130: pname > $o] :
      ( ( collect_pname_o
        @ ^ [X_5: pname > $o] : ( X_5 = A_130 ) )
      = ( insert_pname_o @ A_130 @ bot_bot_pname_o_o ) ) )).

thf(fact_208_singleton__conv,axiom,(
    ! [A_130: hoare_1167836817_state > $o] :
      ( ( collec269976083tate_o
        @ ^ [X_5: hoare_1167836817_state > $o] : ( X_5 = A_130 ) )
      = ( insert999278200tate_o @ A_130 @ bot_bo691907561te_o_o ) ) )).

thf(fact_209_singleton__conv,axiom,(
    ! [A_130: hoare_1167836817_state] :
      ( ( collec1027672124_state
        @ ^ [X_5: hoare_1167836817_state] : ( X_5 = A_130 ) )
      = ( insert2134838167_state @ A_130 @ bot_bo70021908tate_o ) ) )).

thf(fact_210_singleton__conv2,axiom,(
    ! [A_129: pname] :
      ( ( collect_pname @ ( fequal_pname @ A_129 ) )
      = ( insert_pname @ A_129 @ bot_bot_pname_o ) ) )).

thf(fact_211_singleton__conv2,axiom,(
    ! [A_129: pname > $o] :
      ( ( collect_pname_o @ ( fequal_pname_o @ A_129 ) )
      = ( insert_pname_o @ A_129 @ bot_bot_pname_o_o ) ) )).

thf(fact_212_singleton__conv2,axiom,(
    ! [A_129: hoare_1167836817_state > $o] :
      ( ( collec269976083tate_o @ ( fequal1486222077tate_o @ A_129 ) )
      = ( insert999278200tate_o @ A_129 @ bot_bo691907561te_o_o ) ) )).

thf(fact_213_singleton__conv2,axiom,(
    ! [A_129: hoare_1167836817_state] :
      ( ( collec1027672124_state @ ( fequal1831255762_state @ A_129 ) )
      = ( insert2134838167_state @ A_129 @ bot_bo70021908tate_o ) ) )).

thf(fact_214_MGF__lemma1,axiom,(
    ! [C_21: com,G_3: hoare_1167836817_state > $o] :
      ( hoare_1201148605gleton
     => ( ! [X_5: pname] :
            ( ( member_pname @ X_5 @ ( dom_pname_com @ body ) )
           => ( hoare_123228589_state @ G_3 @ ( insert2134838167_state @ ( hoare_Mirabelle_MGT @ ( body_1 @ X_5 ) ) @ bot_bo70021908tate_o ) ) )
       => ( ( wt @ C_21 )
         => ( hoare_123228589_state @ G_3 @ ( insert2134838167_state @ ( hoare_Mirabelle_MGT @ C_21 ) @ bot_bo70021908tate_o ) ) ) ) ) )).

thf(fact_215_WT__bodiesD,axiom,(
    ! [Pn_1: pname,B_42: com] :
      ( wT_bodies
     => ( ( ( body @ Pn_1 )
          = ( some_com @ B_42 ) )
       => ( wt @ B_42 ) ) ) )).

thf(fact_216_imageE,axiom,(
    ! [B_93: pname,F_47: hoare_1167836817_state > pname,A_128: hoare_1167836817_state > $o] :
      ( ( member_pname @ B_93 @ ( image_8178176_pname @ F_47 @ A_128 ) )
     => ~ ! [X_5: hoare_1167836817_state] :
            ( ( B_93
              = ( F_47 @ X_5 ) )
           => ~ ( member2058392318_state @ X_5 @ A_128 ) ) ) )).

thf(fact_217_imageE,axiom,(
    ! [B_93: hoare_1167836817_state,F_47: pname > hoare_1167836817_state,A_128: pname > $o] :
      ( ( member2058392318_state @ B_93 @ ( image_575578384_state @ F_47 @ A_128 ) )
     => ~ ! [X_5: pname] :
            ( ( B_93
              = ( F_47 @ X_5 ) )
           => ~ ( member_pname @ X_5 @ A_128 ) ) ) )).

thf(fact_218_finite__subset__induct,axiom,(
    ! [P_34: ( ( pname > $o ) > $o ) > $o,A_127: ( pname > $o ) > $o,F_46: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ F_46 )
     => ( ( ord_le1205211808me_o_o @ F_46 @ A_127 )
       => ( ( P_34 @ bot_bot_pname_o_o )
         => ( ! [A_122: pname > $o,F_26: ( pname > $o ) > $o] :
                ( ( finite297249702name_o @ F_26 )
               => ( ( member_pname_o @ A_122 @ A_127 )
                 => ( ~ ( member_pname_o @ A_122 @ F_26 )
                   => ( ( P_34 @ F_26 )
                     => ( P_34 @ ( insert_pname_o @ A_122 @ F_26 ) ) ) ) ) )
           => ( P_34 @ F_46 ) ) ) ) ) )).

thf(fact_219_finite__subset__induct,axiom,(
    ! [P_34: ( ( hoare_1167836817_state > $o ) > $o ) > $o,A_127: ( hoare_1167836817_state > $o ) > $o,F_46: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ F_46 )
     => ( ( ord_le741939125te_o_o @ F_46 @ A_127 )
       => ( ( P_34 @ bot_bo691907561te_o_o )
         => ( ! [A_122: hoare_1167836817_state > $o,F_26: ( hoare_1167836817_state > $o ) > $o] :
                ( ( finite1380128977tate_o @ F_26 )
               => ( ( member864234961tate_o @ A_122 @ A_127 )
                 => ( ~ ( member864234961tate_o @ A_122 @ F_26 )
                   => ( ( P_34 @ F_26 )
                     => ( P_34 @ ( insert999278200tate_o @ A_122 @ F_26 ) ) ) ) ) )
           => ( P_34 @ F_46 ) ) ) ) ) )).

thf(fact_220_finite__subset__induct,axiom,(
    ! [P_34: ( pname > $o ) > $o,A_127: pname > $o,F_46: pname > $o] :
      ( ( finite_finite_pname @ F_46 )
     => ( ( ord_less_eq_pname_o @ F_46 @ A_127 )
       => ( ( P_34 @ bot_bot_pname_o )
         => ( ! [A_122: pname,F_26: pname > $o] :
                ( ( finite_finite_pname @ F_26 )
               => ( ( member_pname @ A_122 @ A_127 )
                 => ( ~ ( member_pname @ A_122 @ F_26 )
                   => ( ( P_34 @ F_26 )
                     => ( P_34 @ ( insert_pname @ A_122 @ F_26 ) ) ) ) ) )
           => ( P_34 @ F_46 ) ) ) ) ) )).

thf(fact_221_finite__subset__induct,axiom,(
    ! [P_34: ( hoare_1167836817_state > $o ) > $o,A_127: hoare_1167836817_state > $o,F_46: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ F_46 )
     => ( ( ord_le827224136tate_o @ F_46 @ A_127 )
       => ( ( P_34 @ bot_bo70021908tate_o )
         => ( ! [A_122: hoare_1167836817_state,F_26: hoare_1167836817_state > $o] :
                ( ( finite1084549118_state @ F_26 )
               => ( ( member2058392318_state @ A_122 @ A_127 )
                 => ( ~ ( member2058392318_state @ A_122 @ F_26 )
                   => ( ( P_34 @ F_26 )
                     => ( P_34 @ ( insert2134838167_state @ A_122 @ F_26 ) ) ) ) ) )
           => ( P_34 @ F_46 ) ) ) ) ) )).

thf(fact_222_WTs__elim__cases_I7_J,axiom,(
    ! [P: pname] :
      ( ( wt @ ( body_1 @ P ) )
     => ~ ! [Y_2: com] :
            ( ( body @ P )
           != ( some_com @ Y_2 ) ) ) )).

thf(fact_223_subsetI,axiom,(
    ! [B_92: pname > $o,A_126: pname > $o] :
      ( ! [X_5: pname] :
          ( ( member_pname @ X_5 @ A_126 )
         => ( member_pname @ X_5 @ B_92 ) )
     => ( ord_less_eq_pname_o @ A_126 @ B_92 ) ) )).

thf(fact_224_subsetI,axiom,(
    ! [B_92: hoare_1167836817_state > $o,A_126: hoare_1167836817_state > $o] :
      ( ! [X_5: hoare_1167836817_state] :
          ( ( member2058392318_state @ X_5 @ A_126 )
         => ( member2058392318_state @ X_5 @ B_92 ) )
     => ( ord_le827224136tate_o @ A_126 @ B_92 ) ) )).

thf(fact_225_finite__subset__image,axiom,(
    ! [F_45: ( pname > $o ) > hoare_1167836817_state,A_125: ( pname > $o ) > $o,B_91: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ B_91 )
     => ( ( ord_le827224136tate_o @ B_91 @ ( image_1381916541_state @ F_45 @ A_125 ) )
       => ? [C_47: ( pname > $o ) > $o] :
            ( ( ord_le1205211808me_o_o @ C_47 @ A_125 )
            & ( finite297249702name_o @ C_47 )
            & ( B_91
              = ( image_1381916541_state @ F_45 @ C_47 ) ) ) ) ) )).

thf(fact_226_finite__subset__image,axiom,(
    ! [F_45: ( hoare_1167836817_state > $o ) > hoare_1167836817_state,A_125: ( hoare_1167836817_state > $o ) > $o,B_91: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ B_91 )
     => ( ( ord_le827224136tate_o @ B_91 @ ( image_635813834_state @ F_45 @ A_125 ) )
       => ? [C_47: ( hoare_1167836817_state > $o ) > $o] :
            ( ( ord_le741939125te_o_o @ C_47 @ A_125 )
            & ( finite1380128977tate_o @ C_47 )
            & ( B_91
              = ( image_635813834_state @ F_45 @ C_47 ) ) ) ) ) )).

thf(fact_227_finite__subset__image,axiom,(
    ! [F_45: ( pname > $o ) > pname,A_125: ( pname > $o ) > $o,B_91: pname > $o] :
      ( ( finite_finite_pname @ B_91 )
     => ( ( ord_less_eq_pname_o @ B_91 @ ( image_pname_o_pname @ F_45 @ A_125 ) )
       => ? [C_47: ( pname > $o ) > $o] :
            ( ( ord_le1205211808me_o_o @ C_47 @ A_125 )
            & ( finite297249702name_o @ C_47 )
            & ( B_91
              = ( image_pname_o_pname @ F_45 @ C_47 ) ) ) ) ) )).

thf(fact_228_finite__subset__image,axiom,(
    ! [F_45: ( hoare_1167836817_state > $o ) > pname,A_125: ( hoare_1167836817_state > $o ) > $o,B_91: pname > $o] :
      ( ( finite_finite_pname @ B_91 )
     => ( ( ord_less_eq_pname_o @ B_91 @ ( image_980295115_pname @ F_45 @ A_125 ) )
       => ? [C_47: ( hoare_1167836817_state > $o ) > $o] :
            ( ( ord_le741939125te_o_o @ C_47 @ A_125 )
            & ( finite1380128977tate_o @ C_47 )
            & ( B_91
              = ( image_980295115_pname @ F_45 @ C_47 ) ) ) ) ) )).

thf(fact_229_finite__subset__image,axiom,(
    ! [F_45: hoare_1167836817_state > pname > $o,A_125: hoare_1167836817_state > $o,B_91: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ B_91 )
     => ( ( ord_le1205211808me_o_o @ B_91 @ ( image_2066861949name_o @ F_45 @ A_125 ) )
       => ? [C_47: hoare_1167836817_state > $o] :
            ( ( ord_le827224136tate_o @ C_47 @ A_125 )
            & ( finite1084549118_state @ C_47 )
            & ( B_91
              = ( image_2066861949name_o @ F_45 @ C_47 ) ) ) ) ) )).

thf(fact_230_finite__subset__image,axiom,(
    ! [F_45: hoare_1167836817_state > hoare_1167836817_state > $o,A_125: hoare_1167836817_state > $o,B_91: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ B_91 )
     => ( ( ord_le741939125te_o_o @ B_91 @ ( image_1745649338tate_o @ F_45 @ A_125 ) )
       => ? [C_47: hoare_1167836817_state > $o] :
            ( ( ord_le827224136tate_o @ C_47 @ A_125 )
            & ( finite1084549118_state @ C_47 )
            & ( B_91
              = ( image_1745649338tate_o @ F_45 @ C_47 ) ) ) ) ) )).

thf(fact_231_finite__subset__image,axiom,(
    ! [F_45: hoare_1167836817_state > pname,A_125: hoare_1167836817_state > $o,B_91: pname > $o] :
      ( ( finite_finite_pname @ B_91 )
     => ( ( ord_less_eq_pname_o @ B_91 @ ( image_8178176_pname @ F_45 @ A_125 ) )
       => ? [C_47: hoare_1167836817_state > $o] :
            ( ( ord_le827224136tate_o @ C_47 @ A_125 )
            & ( finite1084549118_state @ C_47 )
            & ( B_91
              = ( image_8178176_pname @ F_45 @ C_47 ) ) ) ) ) )).

thf(fact_232_finite__subset__image,axiom,(
    ! [F_45: pname > pname > $o,A_125: pname > $o,B_91: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ B_91 )
     => ( ( ord_le1205211808me_o_o @ B_91 @ ( image_pname_pname_o @ F_45 @ A_125 ) )
       => ? [C_47: pname > $o] :
            ( ( ord_less_eq_pname_o @ C_47 @ A_125 )
            & ( finite_finite_pname @ C_47 )
            & ( B_91
              = ( image_pname_pname_o @ F_45 @ C_47 ) ) ) ) ) )).

thf(fact_233_finite__subset__image,axiom,(
    ! [F_45: pname > hoare_1167836817_state > $o,A_125: pname > $o,B_91: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ B_91 )
     => ( ( ord_le741939125te_o_o @ B_91 @ ( image_475339327tate_o @ F_45 @ A_125 ) )
       => ? [C_47: pname > $o] :
            ( ( ord_less_eq_pname_o @ C_47 @ A_125 )
            & ( finite_finite_pname @ C_47 )
            & ( B_91
              = ( image_475339327tate_o @ F_45 @ C_47 ) ) ) ) ) )).

thf(fact_234_finite__subset__image,axiom,(
    ! [F_45: pname > pname,A_125: pname > $o,B_91: pname > $o] :
      ( ( finite_finite_pname @ B_91 )
     => ( ( ord_less_eq_pname_o @ B_91 @ ( image_pname_pname @ F_45 @ A_125 ) )
       => ? [C_47: pname > $o] :
            ( ( ord_less_eq_pname_o @ C_47 @ A_125 )
            & ( finite_finite_pname @ C_47 )
            & ( B_91
              = ( image_pname_pname @ F_45 @ C_47 ) ) ) ) ) )).

thf(fact_235_finite__subset__image,axiom,(
    ! [F_45: pname > hoare_1167836817_state,A_125: pname > $o,B_91: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ B_91 )
     => ( ( ord_le827224136tate_o @ B_91 @ ( image_575578384_state @ F_45 @ A_125 ) )
       => ? [C_47: pname > $o] :
            ( ( ord_less_eq_pname_o @ C_47 @ A_125 )
            & ( finite_finite_pname @ C_47 )
            & ( B_91
              = ( image_575578384_state @ F_45 @ C_47 ) ) ) ) ) )).

thf(fact_236_finite__dom__body,axiom,
    ( finite_finite_pname @ ( dom_pname_com @ body ) )).

thf(fact_237_finite__induct,axiom,(
    ! [P_33: ( ( pname > $o ) > $o ) > $o,F_44: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ F_44 )
     => ( ( P_33 @ bot_bot_pname_o_o )
       => ( ! [X_5: pname > $o,F_26: ( pname > $o ) > $o] :
              ( ( finite297249702name_o @ F_26 )
             => ( ~ ( member_pname_o @ X_5 @ F_26 )
               => ( ( P_33 @ F_26 )
                 => ( P_33 @ ( insert_pname_o @ X_5 @ F_26 ) ) ) ) )
         => ( P_33 @ F_44 ) ) ) ) )).

thf(fact_238_finite__induct,axiom,(
    ! [P_33: ( ( hoare_1167836817_state > $o ) > $o ) > $o,F_44: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ F_44 )
     => ( ( P_33 @ bot_bo691907561te_o_o )
       => ( ! [X_5: hoare_1167836817_state > $o,F_26: ( hoare_1167836817_state > $o ) > $o] :
              ( ( finite1380128977tate_o @ F_26 )
             => ( ~ ( member864234961tate_o @ X_5 @ F_26 )
               => ( ( P_33 @ F_26 )
                 => ( P_33 @ ( insert999278200tate_o @ X_5 @ F_26 ) ) ) ) )
         => ( P_33 @ F_44 ) ) ) ) )).

thf(fact_239_finite__induct,axiom,(
    ! [P_33: ( pname > $o ) > $o,F_44: pname > $o] :
      ( ( finite_finite_pname @ F_44 )
     => ( ( P_33 @ bot_bot_pname_o )
       => ( ! [X_5: pname,F_26: pname > $o] :
              ( ( finite_finite_pname @ F_26 )
             => ( ~ ( member_pname @ X_5 @ F_26 )
               => ( ( P_33 @ F_26 )
                 => ( P_33 @ ( insert_pname @ X_5 @ F_26 ) ) ) ) )
         => ( P_33 @ F_44 ) ) ) ) )).

thf(fact_240_finite__induct,axiom,(
    ! [P_33: ( hoare_1167836817_state > $o ) > $o,F_44: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ F_44 )
     => ( ( P_33 @ bot_bo70021908tate_o )
       => ( ! [X_5: hoare_1167836817_state,F_26: hoare_1167836817_state > $o] :
              ( ( finite1084549118_state @ F_26 )
             => ( ~ ( member2058392318_state @ X_5 @ F_26 )
               => ( ( P_33 @ F_26 )
                 => ( P_33 @ ( insert2134838167_state @ X_5 @ F_26 ) ) ) ) )
         => ( P_33 @ F_44 ) ) ) ) )).

thf(fact_241_finite_Osimps,axiom,(
    ! [A_123: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ A_123 )
    <=> ( ( A_123 = bot_bot_pname_o_o )
        | ? [A_124: ( pname > $o ) > $o,A_122: pname > $o] :
            ( ( A_123
              = ( insert_pname_o @ A_122 @ A_124 ) )
            & ( finite297249702name_o @ A_124 ) ) ) ) )).

thf(fact_242_finite_Osimps,axiom,(
    ! [A_123: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ A_123 )
    <=> ( ( A_123 = bot_bo691907561te_o_o )
        | ? [A_124: ( hoare_1167836817_state > $o ) > $o,A_122: hoare_1167836817_state > $o] :
            ( ( A_123
              = ( insert999278200tate_o @ A_122 @ A_124 ) )
            & ( finite1380128977tate_o @ A_124 ) ) ) ) )).

thf(fact_243_finite_Osimps,axiom,(
    ! [A_123: pname > $o] :
      ( ( finite_finite_pname @ A_123 )
    <=> ( ( A_123 = bot_bot_pname_o )
        | ? [A_124: pname > $o,A_122: pname] :
            ( ( A_123
              = ( insert_pname @ A_122 @ A_124 ) )
            & ( finite_finite_pname @ A_124 ) ) ) ) )).

thf(fact_244_finite_Osimps,axiom,(
    ! [A_123: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ A_123 )
    <=> ( ( A_123 = bot_bo70021908tate_o )
        | ? [A_124: hoare_1167836817_state > $o,A_122: hoare_1167836817_state] :
            ( ( A_123
              = ( insert2134838167_state @ A_122 @ A_124 ) )
            & ( finite1084549118_state @ A_124 ) ) ) ) )).

thf(fact_245_pigeonhole__infinite,axiom,(
    ! [F_43: pname > pname > $o,A_121: pname > $o] :
      ( ~ ( finite_finite_pname @ A_121 )
     => ( ( finite297249702name_o @ ( image_pname_pname_o @ F_43 @ A_121 ) )
       => ? [X_5: pname] :
            ( ( member_pname @ X_5 @ A_121 )
            & ~ ( finite_finite_pname
                @ ( collect_pname
                  @ ^ [A_122: pname] :
                      ( (&) @ ( member_pname @ A_122 @ A_121 )
                      @ ( ( F_43 @ A_122 )
                        = ( F_43 @ X_5 ) ) ) ) ) ) ) ) )).

thf(fact_246_pigeonhole__infinite,axiom,(
    ! [F_43: pname > hoare_1167836817_state > $o,A_121: pname > $o] :
      ( ~ ( finite_finite_pname @ A_121 )
     => ( ( finite1380128977tate_o @ ( image_475339327tate_o @ F_43 @ A_121 ) )
       => ? [X_5: pname] :
            ( ( member_pname @ X_5 @ A_121 )
            & ~ ( finite_finite_pname
                @ ( collect_pname
                  @ ^ [A_122: pname] :
                      ( (&) @ ( member_pname @ A_122 @ A_121 )
                      @ ( ( F_43 @ A_122 )
                        = ( F_43 @ X_5 ) ) ) ) ) ) ) ) )).

thf(fact_247_pigeonhole__infinite,axiom,(
    ! [F_43: hoare_1167836817_state > hoare_1167836817_state,A_121: hoare_1167836817_state > $o] :
      ( ~ ( finite1084549118_state @ A_121 )
     => ( ( finite1084549118_state @ ( image_31595733_state @ F_43 @ A_121 ) )
       => ? [X_5: hoare_1167836817_state] :
            ( ( member2058392318_state @ X_5 @ A_121 )
            & ~ ( finite1084549118_state
                @ ( collec1027672124_state
                  @ ^ [A_122: hoare_1167836817_state] :
                      ( (&) @ ( member2058392318_state @ A_122 @ A_121 )
                      @ ( ( F_43 @ A_122 )
                        = ( F_43 @ X_5 ) ) ) ) ) ) ) ) )).

thf(fact_248_pigeonhole__infinite,axiom,(
    ! [F_43: ( pname > $o ) > hoare_1167836817_state,A_121: ( pname > $o ) > $o] :
      ( ~ ( finite297249702name_o @ A_121 )
     => ( ( finite1084549118_state @ ( image_1381916541_state @ F_43 @ A_121 ) )
       => ? [X_5: pname > $o] :
            ( ( member_pname_o @ X_5 @ A_121 )
            & ~ ( finite297249702name_o
                @ ( collect_pname_o
                  @ ^ [A_122: pname > $o] :
                      ( (&) @ ( member_pname_o @ A_122 @ A_121 )
                      @ ( ( F_43 @ A_122 )
                        = ( F_43 @ X_5 ) ) ) ) ) ) ) ) )).

thf(fact_249_pigeonhole__infinite,axiom,(
    ! [F_43: ( hoare_1167836817_state > $o ) > hoare_1167836817_state,A_121: ( hoare_1167836817_state > $o ) > $o] :
      ( ~ ( finite1380128977tate_o @ A_121 )
     => ( ( finite1084549118_state @ ( image_635813834_state @ F_43 @ A_121 ) )
       => ? [X_5: hoare_1167836817_state > $o] :
            ( ( member864234961tate_o @ X_5 @ A_121 )
            & ~ ( finite1380128977tate_o
                @ ( collec269976083tate_o
                  @ ^ [A_122: hoare_1167836817_state > $o] :
                      ( (&) @ ( member864234961tate_o @ A_122 @ A_121 )
                      @ ( ( F_43 @ A_122 )
                        = ( F_43 @ X_5 ) ) ) ) ) ) ) ) )).

thf(fact_250_pigeonhole__infinite,axiom,(
    ! [F_43: pname > pname,A_121: pname > $o] :
      ( ~ ( finite_finite_pname @ A_121 )
     => ( ( finite_finite_pname @ ( image_pname_pname @ F_43 @ A_121 ) )
       => ? [X_5: pname] :
            ( ( member_pname @ X_5 @ A_121 )
            & ~ ( finite_finite_pname
                @ ( collect_pname
                  @ ^ [A_122: pname] :
                      ( (&) @ ( member_pname @ A_122 @ A_121 )
                      @ ( ( F_43 @ A_122 )
                        = ( F_43 @ X_5 ) ) ) ) ) ) ) ) )).

thf(fact_251_pigeonhole__infinite,axiom,(
    ! [F_43: hoare_1167836817_state > pname,A_121: hoare_1167836817_state > $o] :
      ( ~ ( finite1084549118_state @ A_121 )
     => ( ( finite_finite_pname @ ( image_8178176_pname @ F_43 @ A_121 ) )
       => ? [X_5: hoare_1167836817_state] :
            ( ( member2058392318_state @ X_5 @ A_121 )
            & ~ ( finite1084549118_state
                @ ( collec1027672124_state
                  @ ^ [A_122: hoare_1167836817_state] :
                      ( (&) @ ( member2058392318_state @ A_122 @ A_121 )
                      @ ( ( F_43 @ A_122 )
                        = ( F_43 @ X_5 ) ) ) ) ) ) ) ) )).

thf(fact_252_pigeonhole__infinite,axiom,(
    ! [F_43: ( pname > $o ) > pname,A_121: ( pname > $o ) > $o] :
      ( ~ ( finite297249702name_o @ A_121 )
     => ( ( finite_finite_pname @ ( image_pname_o_pname @ F_43 @ A_121 ) )
       => ? [X_5: pname > $o] :
            ( ( member_pname_o @ X_5 @ A_121 )
            & ~ ( finite297249702name_o
                @ ( collect_pname_o
                  @ ^ [A_122: pname > $o] :
                      ( (&) @ ( member_pname_o @ A_122 @ A_121 )
                      @ ( ( F_43 @ A_122 )
                        = ( F_43 @ X_5 ) ) ) ) ) ) ) ) )).

thf(fact_253_pigeonhole__infinite,axiom,(
    ! [F_43: ( hoare_1167836817_state > $o ) > pname,A_121: ( hoare_1167836817_state > $o ) > $o] :
      ( ~ ( finite1380128977tate_o @ A_121 )
     => ( ( finite_finite_pname @ ( image_980295115_pname @ F_43 @ A_121 ) )
       => ? [X_5: hoare_1167836817_state > $o] :
            ( ( member864234961tate_o @ X_5 @ A_121 )
            & ~ ( finite1380128977tate_o
                @ ( collec269976083tate_o
                  @ ^ [A_122: hoare_1167836817_state > $o] :
                      ( (&) @ ( member864234961tate_o @ A_122 @ A_121 )
                      @ ( ( F_43 @ A_122 )
                        = ( F_43 @ X_5 ) ) ) ) ) ) ) ) )).

thf(fact_254_pigeonhole__infinite,axiom,(
    ! [F_43: hoare_1167836817_state > pname > $o,A_121: hoare_1167836817_state > $o] :
      ( ~ ( finite1084549118_state @ A_121 )
     => ( ( finite297249702name_o @ ( image_2066861949name_o @ F_43 @ A_121 ) )
       => ? [X_5: hoare_1167836817_state] :
            ( ( member2058392318_state @ X_5 @ A_121 )
            & ~ ( finite1084549118_state
                @ ( collec1027672124_state
                  @ ^ [A_122: hoare_1167836817_state] :
                      ( (&) @ ( member2058392318_state @ A_122 @ A_121 )
                      @ ( ( F_43 @ A_122 )
                        = ( F_43 @ X_5 ) ) ) ) ) ) ) ) )).

thf(fact_255_pigeonhole__infinite,axiom,(
    ! [F_43: hoare_1167836817_state > hoare_1167836817_state > $o,A_121: hoare_1167836817_state > $o] :
      ( ~ ( finite1084549118_state @ A_121 )
     => ( ( finite1380128977tate_o @ ( image_1745649338tate_o @ F_43 @ A_121 ) )
       => ? [X_5: hoare_1167836817_state] :
            ( ( member2058392318_state @ X_5 @ A_121 )
            & ~ ( finite1084549118_state
                @ ( collec1027672124_state
                  @ ^ [A_122: hoare_1167836817_state] :
                      ( (&) @ ( member2058392318_state @ A_122 @ A_121 )
                      @ ( ( F_43 @ A_122 )
                        = ( F_43 @ X_5 ) ) ) ) ) ) ) ) )).

thf(fact_256_pigeonhole__infinite,axiom,(
    ! [F_43: pname > hoare_1167836817_state,A_121: pname > $o] :
      ( ~ ( finite_finite_pname @ A_121 )
     => ( ( finite1084549118_state @ ( image_575578384_state @ F_43 @ A_121 ) )
       => ? [X_5: pname] :
            ( ( member_pname @ X_5 @ A_121 )
            & ~ ( finite_finite_pname
                @ ( collect_pname
                  @ ^ [A_122: pname] :
                      ( (&) @ ( member_pname @ A_122 @ A_121 )
                      @ ( ( F_43 @ A_122 )
                        = ( F_43 @ X_5 ) ) ) ) ) ) ) ) )).

thf(fact_257_com_Osimps_I6_J,axiom,(
    ! [Pname_1: pname,Pname: pname] :
      ( ( ( body_1 @ Pname_1 )
        = ( body_1 @ Pname ) )
    <=> ( Pname_1 = Pname ) ) )).

thf(fact_258_MGT__Body,axiom,(
    ! [G_3: hoare_1167836817_state > $o,Procs_3: pname > $o] :
      ( ( hoare_123228589_state
        @ ( semila1172322802tate_o @ G_3
          @ ( image_575578384_state
            @ ^ [Pn: pname] :
                ( hoare_Mirabelle_MGT @ ( body_1 @ Pn ) )
            @ Procs_3 ) )
        @ ( image_575578384_state
          @ ^ [Pn: pname] :
              ( hoare_Mirabelle_MGT @ ( the_com @ ( body @ Pn ) ) )
          @ Procs_3 ) )
     => ( ( finite_finite_pname @ Procs_3 )
       => ( hoare_123228589_state @ G_3
          @ ( image_575578384_state
            @ ^ [Pn: pname] :
                ( hoare_Mirabelle_MGT @ ( body_1 @ Pn ) )
            @ Procs_3 ) ) ) ) )).

thf(fact_259_domD,axiom,(
    ! [A_120: pname,M_1: pname > option_com] :
      ( ( member_pname @ A_120 @ ( dom_pname_com @ M_1 ) )
     => ? [B_90: com] :
          ( ( M_1 @ A_120 )
          = ( some_com @ B_90 ) ) ) )).

thf(fact_260_the__elem__eq,axiom,(
    ! [X_84: pname] :
      ( ( the_elem_pname @ ( insert_pname @ X_84 @ bot_bot_pname_o ) )
      = X_84 ) )).

thf(fact_261_the__elem__eq,axiom,(
    ! [X_84: hoare_1167836817_state] :
      ( ( the_el323660082_state @ ( insert2134838167_state @ X_84 @ bot_bo70021908tate_o ) )
      = X_84 ) )).

thf(fact_262_image__subsetI,axiom,(
    ! [F_42: hoare_1167836817_state > pname,B_89: pname > $o,A_119: hoare_1167836817_state > $o] :
      ( ! [X_5: hoare_1167836817_state] :
          ( ( member2058392318_state @ X_5 @ A_119 )
         => ( member_pname @ ( F_42 @ X_5 ) @ B_89 ) )
     => ( ord_less_eq_pname_o @ ( image_8178176_pname @ F_42 @ A_119 ) @ B_89 ) ) )).

thf(fact_263_image__subsetI,axiom,(
    ! [F_42: pname > hoare_1167836817_state,B_89: hoare_1167836817_state > $o,A_119: pname > $o] :
      ( ! [X_5: pname] :
          ( ( member_pname @ X_5 @ A_119 )
         => ( member2058392318_state @ ( F_42 @ X_5 ) @ B_89 ) )
     => ( ord_le827224136tate_o @ ( image_575578384_state @ F_42 @ A_119 ) @ B_89 ) ) )).

thf(fact_264_order__refl,axiom,(
    ! [X_83: pname > $o] :
      ( ord_less_eq_pname_o @ X_83 @ X_83 ) )).

thf(fact_265_order__refl,axiom,(
    ! [X_83: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ X_83 @ X_83 ) )).

thf(fact_266_nonempty__iff,axiom,(
    ! [A_118: pname > $o] :
      ( ( A_118 != bot_bot_pname_o )
    <=> ? [X_5: pname,B_43: pname > $o] :
          ( ( A_118
            = ( insert_pname @ X_5 @ B_43 ) )
          & ~ ( member_pname @ X_5 @ B_43 ) ) ) )).

thf(fact_267_nonempty__iff,axiom,(
    ! [A_118: hoare_1167836817_state > $o] :
      ( ( A_118 != bot_bo70021908tate_o )
    <=> ? [X_5: hoare_1167836817_state,B_43: hoare_1167836817_state > $o] :
          ( ( A_118
            = ( insert2134838167_state @ X_5 @ B_43 ) )
          & ~ ( member2058392318_state @ X_5 @ B_43 ) ) ) )).

thf(fact_268_the_Osimps,axiom,(
    ! [X_82: com] :
      ( ( the_com @ ( some_com @ X_82 ) )
      = X_82 ) )).

thf(fact_269_weak__Body,axiom,(
    ! [G_27: hoare_1167836817_state > $o,P_32: state > state > $o,Pn_4: pname,Q_22: state > state > $o] :
      ( ( hoare_123228589_state @ G_27 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_32 @ ( the_com @ ( body @ Pn_4 ) ) @ Q_22 ) @ bot_bo70021908tate_o ) )
     => ( hoare_123228589_state @ G_27 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_32 @ ( body_1 @ Pn_4 ) @ Q_22 ) @ bot_bo70021908tate_o ) ) ) )).

thf(fact_270_UnCI,axiom,(
    ! [A_117: pname > $o,C_46: pname,B_88: pname > $o] :
      ( ( ~ ( member_pname @ C_46 @ B_88 )
       => ( member_pname @ C_46 @ A_117 ) )
     => ( member_pname @ C_46 @ ( semila1780557381name_o @ A_117 @ B_88 ) ) ) )).

thf(fact_271_UnCI,axiom,(
    ! [A_117: hoare_1167836817_state > $o,C_46: hoare_1167836817_state,B_88: hoare_1167836817_state > $o] :
      ( ( ~ ( member2058392318_state @ C_46 @ B_88 )
       => ( member2058392318_state @ C_46 @ A_117 ) )
     => ( member2058392318_state @ C_46 @ ( semila1172322802tate_o @ A_117 @ B_88 ) ) ) )).

thf(fact_272_UnE,axiom,(
    ! [C_45: pname,A_116: pname > $o,B_87: pname > $o] :
      ( ( member_pname @ C_45 @ ( semila1780557381name_o @ A_116 @ B_87 ) )
     => ( ~ ( member_pname @ C_45 @ A_116 )
       => ( member_pname @ C_45 @ B_87 ) ) ) )).

thf(fact_273_UnE,axiom,(
    ! [C_45: hoare_1167836817_state,A_116: hoare_1167836817_state > $o,B_87: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ C_45 @ ( semila1172322802tate_o @ A_116 @ B_87 ) )
     => ( ~ ( member2058392318_state @ C_45 @ A_116 )
       => ( member2058392318_state @ C_45 @ B_87 ) ) ) )).

thf(fact_274_Collect__disj__eq,axiom,(
    ! [P_31: pname > $o,Q_21: pname > $o] :
      ( ( collect_pname
        @ ^ [X_5: pname] :
            ( (|) @ ( P_31 @ X_5 ) @ ( Q_21 @ X_5 ) ) )
      = ( semila1780557381name_o @ ( collect_pname @ P_31 ) @ ( collect_pname @ Q_21 ) ) ) )).

thf(fact_275_Collect__disj__eq,axiom,(
    ! [P_31: ( pname > $o ) > $o,Q_21: ( pname > $o ) > $o] :
      ( ( collect_pname_o
        @ ^ [X_5: pname > $o] :
            ( (|) @ ( P_31 @ X_5 ) @ ( Q_21 @ X_5 ) ) )
      = ( semila181081674me_o_o @ ( collect_pname_o @ P_31 ) @ ( collect_pname_o @ Q_21 ) ) ) )).

thf(fact_276_Collect__disj__eq,axiom,(
    ! [P_31: ( hoare_1167836817_state > $o ) > $o,Q_21: ( hoare_1167836817_state > $o ) > $o] :
      ( ( collec269976083tate_o
        @ ^ [X_5: hoare_1167836817_state > $o] :
            ( (|) @ ( P_31 @ X_5 ) @ ( Q_21 @ X_5 ) ) )
      = ( semila866907787te_o_o @ ( collec269976083tate_o @ P_31 ) @ ( collec269976083tate_o @ Q_21 ) ) ) )).

thf(fact_277_Collect__disj__eq,axiom,(
    ! [P_31: hoare_1167836817_state > $o,Q_21: hoare_1167836817_state > $o] :
      ( ( collec1027672124_state
        @ ^ [X_5: hoare_1167836817_state] :
            ( (|) @ ( P_31 @ X_5 ) @ ( Q_21 @ X_5 ) ) )
      = ( semila1172322802tate_o @ ( collec1027672124_state @ P_31 ) @ ( collec1027672124_state @ Q_21 ) ) ) )).

thf(fact_278_triple_Oinject,axiom,(
    ! [Fun1_2: state > state > $o,Com_4: com,Fun2_2: state > state > $o,Fun1_1: state > state > $o,Com_3: com,Fun2_1: state > state > $o] :
      ( ( ( hoare_908217195_state @ Fun1_2 @ Com_4 @ Fun2_2 )
        = ( hoare_908217195_state @ Fun1_1 @ Com_3 @ Fun2_1 ) )
    <=> ( ( Fun1_2 = Fun1_1 )
        & ( Com_4 = Com_3 )
        & ( Fun2_2 = Fun2_1 ) ) ) )).

thf(fact_279_UnI2,axiom,(
    ! [A_115: pname > $o,C_44: pname,B_86: pname > $o] :
      ( ( member_pname @ C_44 @ B_86 )
     => ( member_pname @ C_44 @ ( semila1780557381name_o @ A_115 @ B_86 ) ) ) )).

thf(fact_280_UnI2,axiom,(
    ! [A_115: hoare_1167836817_state > $o,C_44: hoare_1167836817_state,B_86: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ C_44 @ B_86 )
     => ( member2058392318_state @ C_44 @ ( semila1172322802tate_o @ A_115 @ B_86 ) ) ) )).

thf(fact_281_UnI1,axiom,(
    ! [B_85: pname > $o,C_43: pname,A_114: pname > $o] :
      ( ( member_pname @ C_43 @ A_114 )
     => ( member_pname @ C_43 @ ( semila1780557381name_o @ A_114 @ B_85 ) ) ) )).

thf(fact_282_UnI1,axiom,(
    ! [B_85: hoare_1167836817_state > $o,C_43: hoare_1167836817_state,A_114: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ C_43 @ A_114 )
     => ( member2058392318_state @ C_43 @ ( semila1172322802tate_o @ A_114 @ B_85 ) ) ) )).

thf(fact_283_ball__Un,axiom,(
    ! [P_30: hoare_1167836817_state > $o,A_113: hoare_1167836817_state > $o,B_84: hoare_1167836817_state > $o] :
      ( ! [X_5: hoare_1167836817_state] :
          ( ( member2058392318_state @ X_5 @ ( semila1172322802tate_o @ A_113 @ B_84 ) )
         => ( P_30 @ X_5 ) )
    <=> ( ! [X_5: hoare_1167836817_state] :
            ( ( member2058392318_state @ X_5 @ A_113 )
           => ( P_30 @ X_5 ) )
        & ! [X_5: hoare_1167836817_state] :
            ( ( member2058392318_state @ X_5 @ B_84 )
           => ( P_30 @ X_5 ) ) ) ) )).

thf(fact_284_bex__Un,axiom,(
    ! [P_29: hoare_1167836817_state > $o,A_112: hoare_1167836817_state > $o,B_83: hoare_1167836817_state > $o] :
      ( ? [X_5: hoare_1167836817_state] :
          ( ( member2058392318_state @ X_5 @ ( semila1172322802tate_o @ A_112 @ B_83 ) )
          & ( P_29 @ X_5 ) )
    <=> ( ? [X_5: hoare_1167836817_state] :
            ( ( member2058392318_state @ X_5 @ A_112 )
            & ( P_29 @ X_5 ) )
        | ? [X_5: hoare_1167836817_state] :
            ( ( member2058392318_state @ X_5 @ B_83 )
            & ( P_29 @ X_5 ) ) ) ) )).

thf(fact_285_Un__assoc,axiom,(
    ! [A_111: hoare_1167836817_state > $o,B_82: hoare_1167836817_state > $o,C_42: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ ( semila1172322802tate_o @ A_111 @ B_82 ) @ C_42 )
      = ( semila1172322802tate_o @ A_111 @ ( semila1172322802tate_o @ B_82 @ C_42 ) ) ) )).

thf(fact_286_Un__iff,axiom,(
    ! [C_41: pname,A_110: pname > $o,B_81: pname > $o] :
      ( ( member_pname @ C_41 @ ( semila1780557381name_o @ A_110 @ B_81 ) )
    <=> ( ( member_pname @ C_41 @ A_110 )
        | ( member_pname @ C_41 @ B_81 ) ) ) )).

thf(fact_287_Un__iff,axiom,(
    ! [C_41: hoare_1167836817_state,A_110: hoare_1167836817_state > $o,B_81: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ C_41 @ ( semila1172322802tate_o @ A_110 @ B_81 ) )
    <=> ( ( member2058392318_state @ C_41 @ A_110 )
        | ( member2058392318_state @ C_41 @ B_81 ) ) ) )).

thf(fact_288_Un__left__commute,axiom,(
    ! [A_109: hoare_1167836817_state > $o,B_80: hoare_1167836817_state > $o,C_40: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ A_109 @ ( semila1172322802tate_o @ B_80 @ C_40 ) )
      = ( semila1172322802tate_o @ B_80 @ ( semila1172322802tate_o @ A_109 @ C_40 ) ) ) )).

thf(fact_289_Un__left__absorb,axiom,(
    ! [A_108: hoare_1167836817_state > $o,B_79: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ A_108 @ ( semila1172322802tate_o @ A_108 @ B_79 ) )
      = ( semila1172322802tate_o @ A_108 @ B_79 ) ) )).

thf(fact_290_Un__commute,axiom,(
    ! [A_107: hoare_1167836817_state > $o,B_78: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ A_107 @ B_78 )
      = ( semila1172322802tate_o @ B_78 @ A_107 ) ) )).

thf(fact_291_Un__def,axiom,(
    ! [A_106: pname > $o,B_77: pname > $o] :
      ( ( semila1780557381name_o @ A_106 @ B_77 )
      = ( collect_pname
        @ ^ [X_5: pname] :
            ( (|) @ ( member_pname @ X_5 @ A_106 ) @ ( member_pname @ X_5 @ B_77 ) ) ) ) )).

thf(fact_292_Un__def,axiom,(
    ! [A_106: ( pname > $o ) > $o,B_77: ( pname > $o ) > $o] :
      ( ( semila181081674me_o_o @ A_106 @ B_77 )
      = ( collect_pname_o
        @ ^ [X_5: pname > $o] :
            ( (|) @ ( member_pname_o @ X_5 @ A_106 ) @ ( member_pname_o @ X_5 @ B_77 ) ) ) ) )).

thf(fact_293_Un__def,axiom,(
    ! [A_106: ( hoare_1167836817_state > $o ) > $o,B_77: ( hoare_1167836817_state > $o ) > $o] :
      ( ( semila866907787te_o_o @ A_106 @ B_77 )
      = ( collec269976083tate_o
        @ ^ [X_5: hoare_1167836817_state > $o] :
            ( (|) @ ( member864234961tate_o @ X_5 @ A_106 ) @ ( member864234961tate_o @ X_5 @ B_77 ) ) ) ) )).

thf(fact_294_Un__def,axiom,(
    ! [A_106: hoare_1167836817_state > $o,B_77: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ A_106 @ B_77 )
      = ( collec1027672124_state
        @ ^ [X_5: hoare_1167836817_state] :
            ( (|) @ ( member2058392318_state @ X_5 @ A_106 ) @ ( member2058392318_state @ X_5 @ B_77 ) ) ) ) )).

thf(fact_295_Un__absorb,axiom,(
    ! [A_105: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ A_105 @ A_105 )
      = A_105 ) )).

thf(fact_296_Un__empty,axiom,(
    ! [A_104: pname > $o,B_76: pname > $o] :
      ( ( ( semila1780557381name_o @ A_104 @ B_76 )
        = bot_bot_pname_o )
    <=> ( ( A_104 = bot_bot_pname_o )
        & ( B_76 = bot_bot_pname_o ) ) ) )).

thf(fact_297_Un__empty,axiom,(
    ! [A_104: hoare_1167836817_state > $o,B_76: hoare_1167836817_state > $o] :
      ( ( ( semila1172322802tate_o @ A_104 @ B_76 )
        = bot_bo70021908tate_o )
    <=> ( ( A_104 = bot_bo70021908tate_o )
        & ( B_76 = bot_bo70021908tate_o ) ) ) )).

thf(fact_298_Un__empty__right,axiom,(
    ! [A_103: pname > $o] :
      ( ( semila1780557381name_o @ A_103 @ bot_bot_pname_o )
      = A_103 ) )).

thf(fact_299_Un__empty__right,axiom,(
    ! [A_103: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ A_103 @ bot_bo70021908tate_o )
      = A_103 ) )).

thf(fact_300_Un__empty__left,axiom,(
    ! [B_75: pname > $o] :
      ( ( semila1780557381name_o @ bot_bot_pname_o @ B_75 )
      = B_75 ) )).

thf(fact_301_Un__empty__left,axiom,(
    ! [B_75: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ bot_bo70021908tate_o @ B_75 )
      = B_75 ) )).

thf(fact_302_finite__UnI,axiom,(
    ! [G_26: ( pname > $o ) > $o,F_41: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ F_41 )
     => ( ( finite297249702name_o @ G_26 )
       => ( finite297249702name_o @ ( semila181081674me_o_o @ F_41 @ G_26 ) ) ) ) )).

thf(fact_303_finite__UnI,axiom,(
    ! [G_26: ( hoare_1167836817_state > $o ) > $o,F_41: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ F_41 )
     => ( ( finite1380128977tate_o @ G_26 )
       => ( finite1380128977tate_o @ ( semila866907787te_o_o @ F_41 @ G_26 ) ) ) ) )).

thf(fact_304_finite__UnI,axiom,(
    ! [G_26: pname > $o,F_41: pname > $o] :
      ( ( finite_finite_pname @ F_41 )
     => ( ( finite_finite_pname @ G_26 )
       => ( finite_finite_pname @ ( semila1780557381name_o @ F_41 @ G_26 ) ) ) ) )).

thf(fact_305_finite__UnI,axiom,(
    ! [G_26: hoare_1167836817_state > $o,F_41: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ F_41 )
     => ( ( finite1084549118_state @ G_26 )
       => ( finite1084549118_state @ ( semila1172322802tate_o @ F_41 @ G_26 ) ) ) ) )).

thf(fact_306_finite__Un,axiom,(
    ! [F_40: ( pname > $o ) > $o,G_25: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ ( semila181081674me_o_o @ F_40 @ G_25 ) )
    <=> ( ( finite297249702name_o @ F_40 )
        & ( finite297249702name_o @ G_25 ) ) ) )).

thf(fact_307_finite__Un,axiom,(
    ! [F_40: ( hoare_1167836817_state > $o ) > $o,G_25: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ ( semila866907787te_o_o @ F_40 @ G_25 ) )
    <=> ( ( finite1380128977tate_o @ F_40 )
        & ( finite1380128977tate_o @ G_25 ) ) ) )).

thf(fact_308_finite__Un,axiom,(
    ! [F_40: pname > $o,G_25: pname > $o] :
      ( ( finite_finite_pname @ ( semila1780557381name_o @ F_40 @ G_25 ) )
    <=> ( ( finite_finite_pname @ F_40 )
        & ( finite_finite_pname @ G_25 ) ) ) )).

thf(fact_309_finite__Un,axiom,(
    ! [F_40: hoare_1167836817_state > $o,G_25: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ ( semila1172322802tate_o @ F_40 @ G_25 ) )
    <=> ( ( finite1084549118_state @ F_40 )
        & ( finite1084549118_state @ G_25 ) ) ) )).

thf(fact_310_Un__insert__left,axiom,(
    ! [A_102: pname,B_74: pname > $o,C_39: pname > $o] :
      ( ( semila1780557381name_o @ ( insert_pname @ A_102 @ B_74 ) @ C_39 )
      = ( insert_pname @ A_102 @ ( semila1780557381name_o @ B_74 @ C_39 ) ) ) )).

thf(fact_311_Un__insert__left,axiom,(
    ! [A_102: hoare_1167836817_state,B_74: hoare_1167836817_state > $o,C_39: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ ( insert2134838167_state @ A_102 @ B_74 ) @ C_39 )
      = ( insert2134838167_state @ A_102 @ ( semila1172322802tate_o @ B_74 @ C_39 ) ) ) )).

thf(fact_312_Un__insert__right,axiom,(
    ! [A_101: pname > $o,A_100: pname,B_73: pname > $o] :
      ( ( semila1780557381name_o @ A_101 @ ( insert_pname @ A_100 @ B_73 ) )
      = ( insert_pname @ A_100 @ ( semila1780557381name_o @ A_101 @ B_73 ) ) ) )).

thf(fact_313_Un__insert__right,axiom,(
    ! [A_101: hoare_1167836817_state > $o,A_100: hoare_1167836817_state,B_73: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ A_101 @ ( insert2134838167_state @ A_100 @ B_73 ) )
      = ( insert2134838167_state @ A_100 @ ( semila1172322802tate_o @ A_101 @ B_73 ) ) ) )).

thf(fact_314_Un__mono,axiom,(
    ! [B_72: pname > $o,D_4: pname > $o,A_99: pname > $o,C_38: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_99 @ C_38 )
     => ( ( ord_less_eq_pname_o @ B_72 @ D_4 )
       => ( ord_less_eq_pname_o @ ( semila1780557381name_o @ A_99 @ B_72 ) @ ( semila1780557381name_o @ C_38 @ D_4 ) ) ) ) )).

thf(fact_315_Un__mono,axiom,(
    ! [B_72: hoare_1167836817_state > $o,D_4: hoare_1167836817_state > $o,A_99: hoare_1167836817_state > $o,C_38: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_99 @ C_38 )
     => ( ( ord_le827224136tate_o @ B_72 @ D_4 )
       => ( ord_le827224136tate_o @ ( semila1172322802tate_o @ A_99 @ B_72 ) @ ( semila1172322802tate_o @ C_38 @ D_4 ) ) ) ) )).

thf(fact_316_Un__least,axiom,(
    ! [B_71: pname > $o,A_98: pname > $o,C_37: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_98 @ C_37 )
     => ( ( ord_less_eq_pname_o @ B_71 @ C_37 )
       => ( ord_less_eq_pname_o @ ( semila1780557381name_o @ A_98 @ B_71 ) @ C_37 ) ) ) )).

thf(fact_317_Un__least,axiom,(
    ! [B_71: hoare_1167836817_state > $o,A_98: hoare_1167836817_state > $o,C_37: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_98 @ C_37 )
     => ( ( ord_le827224136tate_o @ B_71 @ C_37 )
       => ( ord_le827224136tate_o @ ( semila1172322802tate_o @ A_98 @ B_71 ) @ C_37 ) ) ) )).

thf(fact_318_Un__absorb2,axiom,(
    ! [B_70: pname > $o,A_97: pname > $o] :
      ( ( ord_less_eq_pname_o @ B_70 @ A_97 )
     => ( ( semila1780557381name_o @ A_97 @ B_70 )
        = A_97 ) ) )).

thf(fact_319_Un__absorb2,axiom,(
    ! [B_70: hoare_1167836817_state > $o,A_97: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ B_70 @ A_97 )
     => ( ( semila1172322802tate_o @ A_97 @ B_70 )
        = A_97 ) ) )).

thf(fact_320_Un__absorb1,axiom,(
    ! [A_96: pname > $o,B_69: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_96 @ B_69 )
     => ( ( semila1780557381name_o @ A_96 @ B_69 )
        = B_69 ) ) )).

thf(fact_321_Un__absorb1,axiom,(
    ! [A_96: hoare_1167836817_state > $o,B_69: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_96 @ B_69 )
     => ( ( semila1172322802tate_o @ A_96 @ B_69 )
        = B_69 ) ) )).

thf(fact_322_subset__Un__eq,axiom,(
    ! [A_95: pname > $o,B_68: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_95 @ B_68 )
    <=> ( ( semila1780557381name_o @ A_95 @ B_68 )
        = B_68 ) ) )).

thf(fact_323_subset__Un__eq,axiom,(
    ! [A_95: hoare_1167836817_state > $o,B_68: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_95 @ B_68 )
    <=> ( ( semila1172322802tate_o @ A_95 @ B_68 )
        = B_68 ) ) )).

thf(fact_324_Un__upper2,axiom,(
    ! [B_67: pname > $o,A_94: pname > $o] :
      ( ord_less_eq_pname_o @ B_67 @ ( semila1780557381name_o @ A_94 @ B_67 ) ) )).

thf(fact_325_Un__upper2,axiom,(
    ! [B_67: hoare_1167836817_state > $o,A_94: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ B_67 @ ( semila1172322802tate_o @ A_94 @ B_67 ) ) )).

thf(fact_326_Un__upper1,axiom,(
    ! [A_93: pname > $o,B_66: pname > $o] :
      ( ord_less_eq_pname_o @ A_93 @ ( semila1780557381name_o @ A_93 @ B_66 ) ) )).

thf(fact_327_Un__upper1,axiom,(
    ! [A_93: hoare_1167836817_state > $o,B_66: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ A_93 @ ( semila1172322802tate_o @ A_93 @ B_66 ) ) )).

thf(fact_328_image__Un,axiom,(
    ! [F_39: pname > hoare_1167836817_state,A_92: pname > $o,B_65: pname > $o] :
      ( ( image_575578384_state @ F_39 @ ( semila1780557381name_o @ A_92 @ B_65 ) )
      = ( semila1172322802tate_o @ ( image_575578384_state @ F_39 @ A_92 ) @ ( image_575578384_state @ F_39 @ B_65 ) ) ) )).

thf(fact_329_insert__def,axiom,(
    ! [A_91: pname,B_64: pname > $o] :
      ( ( insert_pname @ A_91 @ B_64 )
      = ( semila1780557381name_o
        @ ( collect_pname
          @ ^ [X_5: pname] : ( X_5 = A_91 ) )
        @ B_64 ) ) )).

thf(fact_330_insert__def,axiom,(
    ! [A_91: hoare_1167836817_state,B_64: hoare_1167836817_state > $o] :
      ( ( insert2134838167_state @ A_91 @ B_64 )
      = ( semila1172322802tate_o
        @ ( collec1027672124_state
          @ ^ [X_5: hoare_1167836817_state] : ( X_5 = A_91 ) )
        @ B_64 ) ) )).

thf(fact_331_insert__def,axiom,(
    ! [A_91: pname > $o,B_64: ( pname > $o ) > $o] :
      ( ( insert_pname_o @ A_91 @ B_64 )
      = ( semila181081674me_o_o
        @ ( collect_pname_o
          @ ^ [X_5: pname > $o] : ( X_5 = A_91 ) )
        @ B_64 ) ) )).

thf(fact_332_insert__def,axiom,(
    ! [A_91: hoare_1167836817_state > $o,B_64: ( hoare_1167836817_state > $o ) > $o] :
      ( ( insert999278200tate_o @ A_91 @ B_64 )
      = ( semila866907787te_o_o
        @ ( collec269976083tate_o
          @ ^ [X_5: hoare_1167836817_state > $o] : ( X_5 = A_91 ) )
        @ B_64 ) ) )).

thf(fact_333_insert__is__Un,axiom,(
    ! [A_90: pname,A_89: pname > $o] :
      ( ( insert_pname @ A_90 @ A_89 )
      = ( semila1780557381name_o @ ( insert_pname @ A_90 @ bot_bot_pname_o ) @ A_89 ) ) )).

thf(fact_334_insert__is__Un,axiom,(
    ! [A_90: hoare_1167836817_state,A_89: hoare_1167836817_state > $o] :
      ( ( insert2134838167_state @ A_90 @ A_89 )
      = ( semila1172322802tate_o @ ( insert2134838167_state @ A_90 @ bot_bo70021908tate_o ) @ A_89 ) ) )).

thf(fact_335_hoare__derivs_OBody,axiom,(
    ! [G_24: hoare_1167836817_state > $o,P_28: pname > state > state > $o,Q_20: pname > state > state > $o,Procs_2: pname > $o] :
      ( ( hoare_123228589_state
        @ ( semila1172322802tate_o @ G_24
          @ ( image_575578384_state
            @ ^ [P_24: pname] :
                ( hoare_908217195_state @ ( P_28 @ P_24 ) @ ( body_1 @ P_24 ) @ ( Q_20 @ P_24 ) )
            @ Procs_2 ) )
        @ ( image_575578384_state
          @ ^ [P_24: pname] :
              ( hoare_908217195_state @ ( P_28 @ P_24 ) @ ( the_com @ ( body @ P_24 ) ) @ ( Q_20 @ P_24 ) )
          @ Procs_2 ) )
     => ( hoare_123228589_state @ G_24
        @ ( image_575578384_state
          @ ^ [P_24: pname] :
              ( hoare_908217195_state @ ( P_28 @ P_24 ) @ ( body_1 @ P_24 ) @ ( Q_20 @ P_24 ) )
          @ Procs_2 ) ) ) )).

thf(fact_336_xt1_I6_J,axiom,(
    ! [Z_20: pname > $o,Y_46: pname > $o,X_81: pname > $o] :
      ( ( ord_less_eq_pname_o @ Y_46 @ X_81 )
     => ( ( ord_less_eq_pname_o @ Z_20 @ Y_46 )
       => ( ord_less_eq_pname_o @ Z_20 @ X_81 ) ) ) )).

thf(fact_337_xt1_I6_J,axiom,(
    ! [Z_20: hoare_1167836817_state > $o,Y_46: hoare_1167836817_state > $o,X_81: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ Y_46 @ X_81 )
     => ( ( ord_le827224136tate_o @ Z_20 @ Y_46 )
       => ( ord_le827224136tate_o @ Z_20 @ X_81 ) ) ) )).

thf(fact_338_xt1_I5_J,axiom,(
    ! [Y_45: pname > $o,X_80: pname > $o] :
      ( ( ord_less_eq_pname_o @ Y_45 @ X_80 )
     => ( ( ord_less_eq_pname_o @ X_80 @ Y_45 )
       => ( X_80 = Y_45 ) ) ) )).

thf(fact_339_xt1_I5_J,axiom,(
    ! [Y_45: hoare_1167836817_state > $o,X_80: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ Y_45 @ X_80 )
     => ( ( ord_le827224136tate_o @ X_80 @ Y_45 )
       => ( X_80 = Y_45 ) ) ) )).

thf(fact_340_order__trans,axiom,(
    ! [Z_19: pname > $o,X_79: pname > $o,Y_44: pname > $o] :
      ( ( ord_less_eq_pname_o @ X_79 @ Y_44 )
     => ( ( ord_less_eq_pname_o @ Y_44 @ Z_19 )
       => ( ord_less_eq_pname_o @ X_79 @ Z_19 ) ) ) )).

thf(fact_341_order__trans,axiom,(
    ! [Z_19: hoare_1167836817_state > $o,X_79: hoare_1167836817_state > $o,Y_44: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ X_79 @ Y_44 )
     => ( ( ord_le827224136tate_o @ Y_44 @ Z_19 )
       => ( ord_le827224136tate_o @ X_79 @ Z_19 ) ) ) )).

thf(fact_342_order__antisym,axiom,(
    ! [X_78: pname > $o,Y_43: pname > $o] :
      ( ( ord_less_eq_pname_o @ X_78 @ Y_43 )
     => ( ( ord_less_eq_pname_o @ Y_43 @ X_78 )
       => ( X_78 = Y_43 ) ) ) )).

thf(fact_343_order__antisym,axiom,(
    ! [X_78: hoare_1167836817_state > $o,Y_43: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ X_78 @ Y_43 )
     => ( ( ord_le827224136tate_o @ Y_43 @ X_78 )
       => ( X_78 = Y_43 ) ) ) )).

thf(fact_344_xt1_I4_J,axiom,(
    ! [C_36: pname > $o,B_63: pname > $o,A_88: pname > $o] :
      ( ( ord_less_eq_pname_o @ B_63 @ A_88 )
     => ( ( B_63 = C_36 )
       => ( ord_less_eq_pname_o @ C_36 @ A_88 ) ) ) )).

thf(fact_345_xt1_I4_J,axiom,(
    ! [C_36: hoare_1167836817_state > $o,B_63: hoare_1167836817_state > $o,A_88: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ B_63 @ A_88 )
     => ( ( B_63 = C_36 )
       => ( ord_le827224136tate_o @ C_36 @ A_88 ) ) ) )).

thf(fact_346_ord__le__eq__trans,axiom,(
    ! [C_35: pname > $o,A_87: pname > $o,B_62: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_87 @ B_62 )
     => ( ( B_62 = C_35 )
       => ( ord_less_eq_pname_o @ A_87 @ C_35 ) ) ) )).

thf(fact_347_ord__le__eq__trans,axiom,(
    ! [C_35: hoare_1167836817_state > $o,A_87: hoare_1167836817_state > $o,B_62: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_87 @ B_62 )
     => ( ( B_62 = C_35 )
       => ( ord_le827224136tate_o @ A_87 @ C_35 ) ) ) )).

thf(fact_348_xt1_I3_J,axiom,(
    ! [C_34: pname > $o,A_86: pname > $o,B_61: pname > $o] :
      ( ( A_86 = B_61 )
     => ( ( ord_less_eq_pname_o @ C_34 @ B_61 )
       => ( ord_less_eq_pname_o @ C_34 @ A_86 ) ) ) )).

thf(fact_349_xt1_I3_J,axiom,(
    ! [C_34: hoare_1167836817_state > $o,A_86: hoare_1167836817_state > $o,B_61: hoare_1167836817_state > $o] :
      ( ( A_86 = B_61 )
     => ( ( ord_le827224136tate_o @ C_34 @ B_61 )
       => ( ord_le827224136tate_o @ C_34 @ A_86 ) ) ) )).

thf(fact_350_ord__eq__le__trans,axiom,(
    ! [C_33: pname > $o,A_85: pname > $o,B_60: pname > $o] :
      ( ( A_85 = B_60 )
     => ( ( ord_less_eq_pname_o @ B_60 @ C_33 )
       => ( ord_less_eq_pname_o @ A_85 @ C_33 ) ) ) )).

thf(fact_351_ord__eq__le__trans,axiom,(
    ! [C_33: hoare_1167836817_state > $o,A_85: hoare_1167836817_state > $o,B_60: hoare_1167836817_state > $o] :
      ( ( A_85 = B_60 )
     => ( ( ord_le827224136tate_o @ B_60 @ C_33 )
       => ( ord_le827224136tate_o @ A_85 @ C_33 ) ) ) )).

thf(fact_352_order__antisym__conv,axiom,(
    ! [Y_42: pname > $o,X_77: pname > $o] :
      ( ( ord_less_eq_pname_o @ Y_42 @ X_77 )
     => ( ( ord_less_eq_pname_o @ X_77 @ Y_42 )
      <=> ( X_77 = Y_42 ) ) ) )).

thf(fact_353_order__antisym__conv,axiom,(
    ! [Y_42: hoare_1167836817_state > $o,X_77: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ Y_42 @ X_77 )
     => ( ( ord_le827224136tate_o @ X_77 @ Y_42 )
      <=> ( X_77 = Y_42 ) ) ) )).

thf(fact_354_order__eq__refl,axiom,(
    ! [X_76: pname > $o,Y_41: pname > $o] :
      ( ( X_76 = Y_41 )
     => ( ord_less_eq_pname_o @ X_76 @ Y_41 ) ) )).

thf(fact_355_order__eq__refl,axiom,(
    ! [X_76: hoare_1167836817_state > $o,Y_41: hoare_1167836817_state > $o] :
      ( ( X_76 = Y_41 )
     => ( ord_le827224136tate_o @ X_76 @ Y_41 ) ) )).

thf(fact_356_order__eq__iff,axiom,(
    ! [X_75: pname > $o,Y_40: pname > $o] :
      ( ( X_75 = Y_40 )
    <=> ( ( ord_less_eq_pname_o @ X_75 @ Y_40 )
        & ( ord_less_eq_pname_o @ Y_40 @ X_75 ) ) ) )).

thf(fact_357_order__eq__iff,axiom,(
    ! [X_75: hoare_1167836817_state > $o,Y_40: hoare_1167836817_state > $o] :
      ( ( X_75 = Y_40 )
    <=> ( ( ord_le827224136tate_o @ X_75 @ Y_40 )
        & ( ord_le827224136tate_o @ Y_40 @ X_75 ) ) ) )).

thf(fact_358_option_Oinject,axiom,(
    ! [A_84: com,A_83: com] :
      ( ( ( some_com @ A_84 )
        = ( some_com @ A_83 ) )
    <=> ( A_84 = A_83 ) ) )).

thf(fact_359_constant,axiom,(
    ! [G_23: hoare_1167836817_state > $o,P_27: state > state > $o,C_32: com,Q_19: state > state > $o,C_31: $o] :
      ( ( C_31
       => ( hoare_123228589_state @ G_23 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_27 @ C_32 @ Q_19 ) @ bot_bo70021908tate_o ) ) )
     => ( hoare_123228589_state @ G_23
        @ ( insert2134838167_state
          @ ( hoare_908217195_state
            @ ^ [Z_11: state,S_3: state] :
                ( (&) @ ( P_27 @ Z_11 @ S_3 ) @ C_31 )
            @ C_32
            @ Q_19 )
          @ bot_bo70021908tate_o ) ) ) )).

thf(fact_360_Body1,axiom,(
    ! [Pn_3: pname,G_22: hoare_1167836817_state > $o,P_26: pname > state > state > $o,Q_18: pname > state > state > $o,Procs_1: pname > $o] :
      ( ( hoare_123228589_state
        @ ( semila1172322802tate_o @ G_22
          @ ( image_575578384_state
            @ ^ [P_24: pname] :
                ( hoare_908217195_state @ ( P_26 @ P_24 ) @ ( body_1 @ P_24 ) @ ( Q_18 @ P_24 ) )
            @ Procs_1 ) )
        @ ( image_575578384_state
          @ ^ [P_24: pname] :
              ( hoare_908217195_state @ ( P_26 @ P_24 ) @ ( the_com @ ( body @ P_24 ) ) @ ( Q_18 @ P_24 ) )
          @ Procs_1 ) )
     => ( ( member_pname @ Pn_3 @ Procs_1 )
       => ( hoare_123228589_state @ G_22 @ ( insert2134838167_state @ ( hoare_908217195_state @ ( P_26 @ Pn_3 ) @ ( body_1 @ Pn_3 ) @ ( Q_18 @ Pn_3 ) ) @ bot_bo70021908tate_o ) ) ) ) )).

thf(fact_361_le__bot,axiom,(
    ! [A_82: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_82 @ bot_bot_pname_o )
     => ( A_82 = bot_bot_pname_o ) ) )).

thf(fact_362_le__bot,axiom,(
    ! [A_82: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_82 @ bot_bo70021908tate_o )
     => ( A_82 = bot_bo70021908tate_o ) ) )).

thf(fact_363_bot__unique,axiom,(
    ! [A_81: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_81 @ bot_bot_pname_o )
    <=> ( A_81 = bot_bot_pname_o ) ) )).

thf(fact_364_bot__unique,axiom,(
    ! [A_81: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_81 @ bot_bo70021908tate_o )
    <=> ( A_81 = bot_bo70021908tate_o ) ) )).

thf(fact_365_bot__least,axiom,(
    ! [A_80: pname > $o] :
      ( ord_less_eq_pname_o @ bot_bot_pname_o @ A_80 ) )).

thf(fact_366_bot__least,axiom,(
    ! [A_80: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ bot_bo70021908tate_o @ A_80 ) )).

thf(fact_367_le__funE,axiom,(
    ! [X_74: pname,F_38: pname > $o,G_21: pname > $o] :
      ( ( ord_less_eq_pname_o @ F_38 @ G_21 )
     => ( ord_less_eq_o @ ( F_38 @ X_74 ) @ ( G_21 @ X_74 ) ) ) )).

thf(fact_368_le__funE,axiom,(
    ! [X_74: hoare_1167836817_state,F_38: hoare_1167836817_state > $o,G_21: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ F_38 @ G_21 )
     => ( ord_less_eq_o @ ( F_38 @ X_74 ) @ ( G_21 @ X_74 ) ) ) )).

thf(fact_369_le__funD,axiom,(
    ! [X_73: pname,F_37: pname > $o,G_20: pname > $o] :
      ( ( ord_less_eq_pname_o @ F_37 @ G_20 )
     => ( ord_less_eq_o @ ( F_37 @ X_73 ) @ ( G_20 @ X_73 ) ) ) )).

thf(fact_370_le__funD,axiom,(
    ! [X_73: hoare_1167836817_state,F_37: hoare_1167836817_state > $o,G_20: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ F_37 @ G_20 )
     => ( ord_less_eq_o @ ( F_37 @ X_73 ) @ ( G_20 @ X_73 ) ) ) )).

thf(fact_371_le__fun__def,axiom,(
    ! [F_36: pname > $o,G_19: pname > $o] :
      ( ( ord_less_eq_pname_o @ F_36 @ G_19 )
    <=> ! [X_5: pname] :
          ( ord_less_eq_o @ ( F_36 @ X_5 ) @ ( G_19 @ X_5 ) ) ) )).

thf(fact_372_le__fun__def,axiom,(
    ! [F_36: hoare_1167836817_state > $o,G_19: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ F_36 @ G_19 )
    <=> ! [X_5: hoare_1167836817_state] :
          ( ord_less_eq_o @ ( F_36 @ X_5 ) @ ( G_19 @ X_5 ) ) ) )).

thf(fact_373_bot__apply,axiom,(
    ! [X_72: pname] :
      ( ( bot_bot_pname_o @ X_72 )
    <=> bot_bot_o ) )).

thf(fact_374_bot__apply,axiom,(
    ! [X_72: hoare_1167836817_state] :
      ( ( bot_bo70021908tate_o @ X_72 )
    <=> bot_bot_o ) )).

thf(fact_375_bot__fun__def,axiom,(
    ! [X_5: pname] :
      ( ( bot_bot_pname_o @ X_5 )
    <=> bot_bot_o ) )).

thf(fact_376_bot__fun__def,axiom,(
    ! [X_5: hoare_1167836817_state] :
      ( ( bot_bo70021908tate_o @ X_5 )
    <=> bot_bot_o ) )).

thf(fact_377_BodyN,axiom,(
    ! [P_25: state > state > $o,Pn_2: pname,Q_17: state > state > $o,G_18: hoare_1167836817_state > $o] :
      ( ( hoare_123228589_state @ ( insert2134838167_state @ ( hoare_908217195_state @ P_25 @ ( body_1 @ Pn_2 ) @ Q_17 ) @ G_18 ) @ ( insert2134838167_state @ ( hoare_908217195_state @ P_25 @ ( the_com @ ( body @ Pn_2 ) ) @ Q_17 ) @ bot_bo70021908tate_o ) )
     => ( hoare_123228589_state @ G_18 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_25 @ ( body_1 @ Pn_2 ) @ Q_17 ) @ bot_bo70021908tate_o ) ) ) )).

thf(fact_378_finite__pointwise,axiom,(
    ! [P_23: pname > state > state > $o,Q_16: pname > state > state > $o,G_17: hoare_1167836817_state > $o,P_22: pname > state > state > $o,C0_1: pname > com,Q_15: pname > state > state > $o,U_1: pname > $o] :
      ( ( finite_finite_pname @ U_1 )
     => ( ! [P_24: pname] :
            ( ( hoare_123228589_state @ G_17 @ ( insert2134838167_state @ ( hoare_908217195_state @ ( P_22 @ P_24 ) @ ( C0_1 @ P_24 ) @ ( Q_15 @ P_24 ) ) @ bot_bo70021908tate_o ) )
           => ( hoare_123228589_state @ G_17 @ ( insert2134838167_state @ ( hoare_908217195_state @ ( P_23 @ P_24 ) @ ( C0_1 @ P_24 ) @ ( Q_16 @ P_24 ) ) @ bot_bo70021908tate_o ) ) )
       => ( ( hoare_123228589_state @ G_17
            @ ( image_575578384_state
              @ ^ [P_24: pname] :
                  ( hoare_908217195_state @ ( P_22 @ P_24 ) @ ( C0_1 @ P_24 ) @ ( Q_15 @ P_24 ) )
              @ U_1 ) )
         => ( hoare_123228589_state @ G_17
            @ ( image_575578384_state
              @ ^ [P_24: pname] :
                  ( hoare_908217195_state @ ( P_23 @ P_24 ) @ ( C0_1 @ P_24 ) @ ( Q_16 @ P_24 ) )
              @ U_1 ) ) ) ) ) )).

thf(fact_379_escape,axiom,(
    ! [G_16: hoare_1167836817_state > $o,C_30: com,Q_14: state > state > $o,P_21: state > state > $o] :
      ( ! [Z_11: state,S_3: state] :
          ( ( P_21 @ Z_11 @ S_3 )
         => ( hoare_123228589_state @ G_16
            @ ( insert2134838167_state
              @ ( hoare_908217195_state
                @ ^ [Za: state,S_4: state] : ( S_4 = S_3 )
                @ C_30
                @ ^ [Z_18: state] :
                    ( Q_14 @ Z_11 ) )
              @ bot_bo70021908tate_o ) ) )
     => ( hoare_123228589_state @ G_16 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_21 @ C_30 @ Q_14 ) @ bot_bo70021908tate_o ) ) ) )).

thf(fact_380_Body__sound__lemma,axiom,(
    ! [G_15: hoare_1167836817_state > $o,P_20: pname > state > state > $o,Q_13: pname > state > state > $o,Procs: pname > $o] :
      ( ( hoare_529639851_state
        @ ( semila1172322802tate_o @ G_15
          @ ( image_575578384_state
            @ ^ [Pn: pname] :
                ( hoare_908217195_state @ ( P_20 @ Pn ) @ ( body_1 @ Pn ) @ ( Q_13 @ Pn ) )
            @ Procs ) )
        @ ( image_575578384_state
          @ ^ [Pn: pname] :
              ( hoare_908217195_state @ ( P_20 @ Pn ) @ ( the_com @ ( body @ Pn ) ) @ ( Q_13 @ Pn ) )
          @ Procs ) )
     => ( hoare_529639851_state @ G_15
        @ ( image_575578384_state
          @ ^ [Pn: pname] :
              ( hoare_908217195_state @ ( P_20 @ Pn ) @ ( body_1 @ Pn ) @ ( Q_13 @ Pn ) )
          @ Procs ) ) ) )).

thf(fact_381_conseq1,axiom,(
    ! [P_19: state > state > $o,G_14: hoare_1167836817_state > $o,P_18: state > state > $o,C_29: com,Q_12: state > state > $o] :
      ( ( hoare_123228589_state @ G_14 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_18 @ C_29 @ Q_12 ) @ bot_bo70021908tate_o ) )
     => ( ! [Z_11: state,S_3: state] :
            ( ( P_19 @ Z_11 @ S_3 )
           => ( P_18 @ Z_11 @ S_3 ) )
       => ( hoare_123228589_state @ G_14 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_19 @ C_29 @ Q_12 ) @ bot_bo70021908tate_o ) ) ) ) )).

thf(fact_382_conseq2,axiom,(
    ! [Q_11: state > state > $o,G_13: hoare_1167836817_state > $o,P_17: state > state > $o,C_28: com,Q_10: state > state > $o] :
      ( ( hoare_123228589_state @ G_13 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_17 @ C_28 @ Q_10 ) @ bot_bo70021908tate_o ) )
     => ( ! [Z_11: state,S_3: state] :
            ( ( Q_10 @ Z_11 @ S_3 )
           => ( Q_11 @ Z_11 @ S_3 ) )
       => ( hoare_123228589_state @ G_13 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_17 @ C_28 @ Q_11 ) @ bot_bo70021908tate_o ) ) ) ) )).

thf(fact_383_MGF__complete,axiom,(
    ! [P: state > state > $o,Q_9: state > state > $o,C_21: com] :
      ( ( hoare_123228589_state @ bot_bo70021908tate_o @ ( insert2134838167_state @ ( hoare_Mirabelle_MGT @ C_21 ) @ bot_bo70021908tate_o ) )
     => ( ( hoare_529639851_state @ bot_bo70021908tate_o @ ( insert2134838167_state @ ( hoare_908217195_state @ P @ C_21 @ Q_9 ) @ bot_bo70021908tate_o ) )
       => ( hoare_123228589_state @ bot_bo70021908tate_o @ ( insert2134838167_state @ ( hoare_908217195_state @ P @ C_21 @ Q_9 ) @ bot_bo70021908tate_o ) ) ) ) )).

thf(fact_384_sup1E,axiom,(
    ! [A_79: hoare_1167836817_state > $o,B_59: hoare_1167836817_state > $o,X_71: hoare_1167836817_state] :
      ( ( semila1172322802tate_o @ A_79 @ B_59 @ X_71 )
     => ( ~ ( A_79 @ X_71 )
       => ( B_59 @ X_71 ) ) ) )).

thf(fact_385_sup1CI,axiom,(
    ! [A_78: hoare_1167836817_state > $o,B_58: hoare_1167836817_state > $o,X_70: hoare_1167836817_state] :
      ( ( ~ ( B_58 @ X_70 )
       => ( A_78 @ X_70 ) )
     => ( semila1172322802tate_o @ A_78 @ B_58 @ X_70 ) ) )).

thf(fact_386_hoare__sound,axiom,(
    ! [G_12: hoare_1167836817_state > $o,Ts: hoare_1167836817_state > $o] :
      ( ( hoare_123228589_state @ G_12 @ Ts )
     => ( hoare_529639851_state @ G_12 @ Ts ) ) )).

thf(fact_387_bot__empty__eq,axiom,(
    ! [X_5: pname] :
      ( ( bot_bot_pname_o @ X_5 )
    <=> ( member_pname @ X_5 @ bot_bot_pname_o ) ) )).

thf(fact_388_bot__empty__eq,axiom,(
    ! [X_5: hoare_1167836817_state] :
      ( ( bot_bo70021908tate_o @ X_5 )
    <=> ( member2058392318_state @ X_5 @ bot_bo70021908tate_o ) ) )).

thf(fact_389_rev__predicate1D,axiom,(
    ! [Q_8: pname > $o,P_16: pname > $o,X_69: pname] :
      ( ( P_16 @ X_69 )
     => ( ( ord_less_eq_pname_o @ P_16 @ Q_8 )
       => ( Q_8 @ X_69 ) ) ) )).

thf(fact_390_rev__predicate1D,axiom,(
    ! [Q_8: hoare_1167836817_state > $o,P_16: hoare_1167836817_state > $o,X_69: hoare_1167836817_state] :
      ( ( P_16 @ X_69 )
     => ( ( ord_le827224136tate_o @ P_16 @ Q_8 )
       => ( Q_8 @ X_69 ) ) ) )).

thf(fact_391_predicate1D,axiom,(
    ! [X_68: pname,P_15: pname > $o,Q_7: pname > $o] :
      ( ( ord_less_eq_pname_o @ P_15 @ Q_7 )
     => ( ( P_15 @ X_68 )
       => ( Q_7 @ X_68 ) ) ) )).

thf(fact_392_predicate1D,axiom,(
    ! [X_68: hoare_1167836817_state,P_15: hoare_1167836817_state > $o,Q_7: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ P_15 @ Q_7 )
     => ( ( P_15 @ X_68 )
       => ( Q_7 @ X_68 ) ) ) )).

thf(fact_393_sup1I2,axiom,(
    ! [A_77: hoare_1167836817_state > $o,B_57: hoare_1167836817_state > $o,X_67: hoare_1167836817_state] :
      ( ( B_57 @ X_67 )
     => ( semila1172322802tate_o @ A_77 @ B_57 @ X_67 ) ) )).

thf(fact_394_sup1I1,axiom,(
    ! [B_56: hoare_1167836817_state > $o,A_76: hoare_1167836817_state > $o,X_66: hoare_1167836817_state] :
      ( ( A_76 @ X_66 )
     => ( semila1172322802tate_o @ A_76 @ B_56 @ X_66 ) ) )).

thf(fact_395_pred__subset__eq,axiom,(
    ! [R_3: pname > $o,S_6: pname > $o] :
      ( ( ord_less_eq_pname_o
        @ ^ [X_5: pname] :
            ( member_pname @ X_5 @ R_3 )
        @ ^ [X_5: pname] :
            ( member_pname @ X_5 @ S_6 ) )
    <=> ( ord_less_eq_pname_o @ R_3 @ S_6 ) ) )).

thf(fact_396_pred__subset__eq,axiom,(
    ! [R_3: hoare_1167836817_state > $o,S_6: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o
        @ ^ [X_5: hoare_1167836817_state] :
            ( member2058392318_state @ X_5 @ R_3 )
        @ ^ [X_5: hoare_1167836817_state] :
            ( member2058392318_state @ X_5 @ S_6 ) )
    <=> ( ord_le827224136tate_o @ R_3 @ S_6 ) ) )).

thf(fact_397_sup__Un__eq,axiom,(
    ! [R_2: pname > $o,S_5: pname > $o,X_5: pname] :
      ( ( semila1780557381name_o
        @ ^ [Y_2: pname] :
            ( member_pname @ Y_2 @ R_2 )
        @ ^ [Y_2: pname] :
            ( member_pname @ Y_2 @ S_5 )
        @ X_5 )
    <=> ( member_pname @ X_5 @ ( semila1780557381name_o @ R_2 @ S_5 ) ) ) )).

thf(fact_398_sup__Un__eq,axiom,(
    ! [R_2: hoare_1167836817_state > $o,S_5: hoare_1167836817_state > $o,X_5: hoare_1167836817_state] :
      ( ( semila1172322802tate_o
        @ ^ [Y_2: hoare_1167836817_state] :
            ( member2058392318_state @ Y_2 @ R_2 )
        @ ^ [Y_2: hoare_1167836817_state] :
            ( member2058392318_state @ Y_2 @ S_5 )
        @ X_5 )
    <=> ( member2058392318_state @ X_5 @ ( semila1172322802tate_o @ R_2 @ S_5 ) ) ) )).

thf(fact_399_conseq12,axiom,(
    ! [Q_6: state > state > $o,P_14: state > state > $o,G_11: hoare_1167836817_state > $o,P_13: state > state > $o,C_27: com,Q_5: state > state > $o] :
      ( ( hoare_123228589_state @ G_11 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_13 @ C_27 @ Q_5 ) @ bot_bo70021908tate_o ) )
     => ( ! [Z_11: state,S_3: state] :
            ( ( P_14 @ Z_11 @ S_3 )
           => ! [S_4: state] :
                ( ! [Z_18: state] :
                    ( ( P_13 @ Z_18 @ S_3 )
                   => ( Q_5 @ Z_18 @ S_4 ) )
               => ( Q_6 @ Z_11 @ S_4 ) ) )
       => ( hoare_123228589_state @ G_11 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_14 @ C_27 @ Q_6 ) @ bot_bo70021908tate_o ) ) ) ) )).

thf(fact_400_le__funI,axiom,(
    ! [F_35: pname > $o,G_10: pname > $o] :
      ( ! [X_5: pname] :
          ( ord_less_eq_o @ ( F_35 @ X_5 ) @ ( G_10 @ X_5 ) )
     => ( ord_less_eq_pname_o @ F_35 @ G_10 ) ) )).

thf(fact_401_le__funI,axiom,(
    ! [F_35: hoare_1167836817_state > $o,G_10: hoare_1167836817_state > $o] :
      ( ! [X_5: hoare_1167836817_state] :
          ( ord_less_eq_o @ ( F_35 @ X_5 ) @ ( G_10 @ X_5 ) )
     => ( ord_le827224136tate_o @ F_35 @ G_10 ) ) )).

thf(fact_402_Option_Oset_Osimps_I2_J,axiom,(
    ! [X_65: pname] :
      ( ( set_pname @ ( some_pname @ X_65 ) )
      = ( insert_pname @ X_65 @ bot_bot_pname_o ) ) )).

thf(fact_403_Option_Oset_Osimps_I2_J,axiom,(
    ! [X_65: hoare_1167836817_state] :
      ( ( set_Ho2131684873_state @ ( some_H1433514562_state @ X_65 ) )
      = ( insert2134838167_state @ X_65 @ bot_bo70021908tate_o ) ) )).

thf(fact_404_Option_Oset_Osimps_I2_J,axiom,(
    ! [X_65: com] :
      ( ( set_com @ ( some_com @ X_65 ) )
      = ( insert_com @ X_65 @ bot_bot_com_o ) ) )).

thf(fact_405_elem__set,axiom,(
    ! [X_64: pname,Xo: option_pname] :
      ( ( member_pname @ X_64 @ ( set_pname @ Xo ) )
    <=> ( Xo
        = ( some_pname @ X_64 ) ) ) )).

thf(fact_406_elem__set,axiom,(
    ! [X_64: hoare_1167836817_state,Xo: option1574264306_state] :
      ( ( member2058392318_state @ X_64 @ ( set_Ho2131684873_state @ Xo ) )
    <=> ( Xo
        = ( some_H1433514562_state @ X_64 ) ) ) )).

thf(fact_407_elem__set,axiom,(
    ! [X_64: com,Xo: option_com] :
      ( ( member_com @ X_64 @ ( set_com @ Xo ) )
    <=> ( Xo
        = ( some_com @ X_64 ) ) ) )).

thf(fact_408_ospec,axiom,(
    ! [X_63: com,P_12: com > $o,A_75: option_com] :
      ( ! [X_5: com] :
          ( ( member_com @ X_5 @ ( set_com @ A_75 ) )
         => ( P_12 @ X_5 ) )
     => ( ( A_75
          = ( some_com @ X_63 ) )
       => ( P_12 @ X_63 ) ) ) )).

thf(fact_409_sup__fun__def,axiom,(
    ! [F_34: hoare_1167836817_state > $o,G_9: hoare_1167836817_state > $o,X_5: hoare_1167836817_state] :
      ( ( semila1172322802tate_o @ F_34 @ G_9 @ X_5 )
    <=> ( semila10642723_sup_o @ ( F_34 @ X_5 ) @ ( G_9 @ X_5 ) ) ) )).

thf(fact_410_sup__apply,axiom,(
    ! [F_33: hoare_1167836817_state > $o,G_8: hoare_1167836817_state > $o,X_62: hoare_1167836817_state] :
      ( ( semila1172322802tate_o @ F_33 @ G_8 @ X_62 )
    <=> ( semila10642723_sup_o @ ( F_33 @ X_62 ) @ ( G_8 @ X_62 ) ) ) )).

thf(fact_411_single__stateE,axiom,
    ( hoare_1201148605gleton
   => ! [T_1: state] :
        ~ ! [S_3: state] : ( S_3 = T_1 ) )).

thf(fact_412_state__not__singleton__def,axiom,
    ( hoare_1201148605gleton
  <=> ? [S_3: state,T_1: state] : ( S_3 != T_1 ) )).

thf(fact_413_sup__assoc,axiom,(
    ! [X_61: hoare_1167836817_state > $o,Y_39: hoare_1167836817_state > $o,Z_17: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ ( semila1172322802tate_o @ X_61 @ Y_39 ) @ Z_17 )
      = ( semila1172322802tate_o @ X_61 @ ( semila1172322802tate_o @ Y_39 @ Z_17 ) ) ) )).

thf(fact_414_inf__sup__aci_I6_J,axiom,(
    ! [X_60: hoare_1167836817_state > $o,Y_38: hoare_1167836817_state > $o,Z_16: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ ( semila1172322802tate_o @ X_60 @ Y_38 ) @ Z_16 )
      = ( semila1172322802tate_o @ X_60 @ ( semila1172322802tate_o @ Y_38 @ Z_16 ) ) ) )).

thf(fact_415_sup_Oassoc,axiom,(
    ! [A_74: hoare_1167836817_state > $o,B_55: hoare_1167836817_state > $o,C_26: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ ( semila1172322802tate_o @ A_74 @ B_55 ) @ C_26 )
      = ( semila1172322802tate_o @ A_74 @ ( semila1172322802tate_o @ B_55 @ C_26 ) ) ) )).

thf(fact_416_sup__left__commute,axiom,(
    ! [X_59: hoare_1167836817_state > $o,Y_37: hoare_1167836817_state > $o,Z_15: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ X_59 @ ( semila1172322802tate_o @ Y_37 @ Z_15 ) )
      = ( semila1172322802tate_o @ Y_37 @ ( semila1172322802tate_o @ X_59 @ Z_15 ) ) ) )).

thf(fact_417_inf__sup__aci_I7_J,axiom,(
    ! [X_58: hoare_1167836817_state > $o,Y_36: hoare_1167836817_state > $o,Z_14: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ X_58 @ ( semila1172322802tate_o @ Y_36 @ Z_14 ) )
      = ( semila1172322802tate_o @ Y_36 @ ( semila1172322802tate_o @ X_58 @ Z_14 ) ) ) )).

thf(fact_418_sup_Oleft__commute,axiom,(
    ! [B_54: hoare_1167836817_state > $o,A_73: hoare_1167836817_state > $o,C_25: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ B_54 @ ( semila1172322802tate_o @ A_73 @ C_25 ) )
      = ( semila1172322802tate_o @ A_73 @ ( semila1172322802tate_o @ B_54 @ C_25 ) ) ) )).

thf(fact_419_sup__left__idem,axiom,(
    ! [X_57: hoare_1167836817_state > $o,Y_35: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ X_57 @ ( semila1172322802tate_o @ X_57 @ Y_35 ) )
      = ( semila1172322802tate_o @ X_57 @ Y_35 ) ) )).

thf(fact_420_inf__sup__aci_I8_J,axiom,(
    ! [X_56: hoare_1167836817_state > $o,Y_34: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ X_56 @ ( semila1172322802tate_o @ X_56 @ Y_34 ) )
      = ( semila1172322802tate_o @ X_56 @ Y_34 ) ) )).

thf(fact_421_sup_Oleft__idem,axiom,(
    ! [A_72: hoare_1167836817_state > $o,B_53: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ A_72 @ ( semila1172322802tate_o @ A_72 @ B_53 ) )
      = ( semila1172322802tate_o @ A_72 @ B_53 ) ) )).

thf(fact_422_sup__commute,axiom,(
    ! [X_55: hoare_1167836817_state > $o,Y_33: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ X_55 @ Y_33 )
      = ( semila1172322802tate_o @ Y_33 @ X_55 ) ) )).

thf(fact_423_inf__sup__aci_I5_J,axiom,(
    ! [X_54: hoare_1167836817_state > $o,Y_32: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ X_54 @ Y_32 )
      = ( semila1172322802tate_o @ Y_32 @ X_54 ) ) )).

thf(fact_424_sup_Ocommute,axiom,(
    ! [A_71: hoare_1167836817_state > $o,B_52: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ A_71 @ B_52 )
      = ( semila1172322802tate_o @ B_52 @ A_71 ) ) )).

thf(fact_425_sup__idem,axiom,(
    ! [X_53: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ X_53 @ X_53 )
      = X_53 ) )).

thf(fact_426_sup_Oidem,axiom,(
    ! [A_70: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ A_70 @ A_70 )
      = A_70 ) )).

thf(fact_427_le__supE,axiom,(
    ! [A_69: pname > $o,B_51: pname > $o,X_52: pname > $o] :
      ( ( ord_less_eq_pname_o @ ( semila1780557381name_o @ A_69 @ B_51 ) @ X_52 )
     => ~ ( ( ord_less_eq_pname_o @ A_69 @ X_52 )
         => ~ ( ord_less_eq_pname_o @ B_51 @ X_52 ) ) ) )).

thf(fact_428_le__supE,axiom,(
    ! [A_69: hoare_1167836817_state > $o,B_51: hoare_1167836817_state > $o,X_52: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ ( semila1172322802tate_o @ A_69 @ B_51 ) @ X_52 )
     => ~ ( ( ord_le827224136tate_o @ A_69 @ X_52 )
         => ~ ( ord_le827224136tate_o @ B_51 @ X_52 ) ) ) )).

thf(fact_429_sup__mono,axiom,(
    ! [B_50: pname > $o,D_3: pname > $o,A_68: pname > $o,C_24: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_68 @ C_24 )
     => ( ( ord_less_eq_pname_o @ B_50 @ D_3 )
       => ( ord_less_eq_pname_o @ ( semila1780557381name_o @ A_68 @ B_50 ) @ ( semila1780557381name_o @ C_24 @ D_3 ) ) ) ) )).

thf(fact_430_sup__mono,axiom,(
    ! [B_50: hoare_1167836817_state > $o,D_3: hoare_1167836817_state > $o,A_68: hoare_1167836817_state > $o,C_24: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_68 @ C_24 )
     => ( ( ord_le827224136tate_o @ B_50 @ D_3 )
       => ( ord_le827224136tate_o @ ( semila1172322802tate_o @ A_68 @ B_50 ) @ ( semila1172322802tate_o @ C_24 @ D_3 ) ) ) ) )).

thf(fact_431_sup__least,axiom,(
    ! [Z_13: pname > $o,Y_31: pname > $o,X_51: pname > $o] :
      ( ( ord_less_eq_pname_o @ Y_31 @ X_51 )
     => ( ( ord_less_eq_pname_o @ Z_13 @ X_51 )
       => ( ord_less_eq_pname_o @ ( semila1780557381name_o @ Y_31 @ Z_13 ) @ X_51 ) ) ) )).

thf(fact_432_sup__least,axiom,(
    ! [Z_13: hoare_1167836817_state > $o,Y_31: hoare_1167836817_state > $o,X_51: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ Y_31 @ X_51 )
     => ( ( ord_le827224136tate_o @ Z_13 @ X_51 )
       => ( ord_le827224136tate_o @ ( semila1172322802tate_o @ Y_31 @ Z_13 ) @ X_51 ) ) ) )).

thf(fact_433_le__supI,axiom,(
    ! [B_49: pname > $o,A_67: pname > $o,X_50: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_67 @ X_50 )
     => ( ( ord_less_eq_pname_o @ B_49 @ X_50 )
       => ( ord_less_eq_pname_o @ ( semila1780557381name_o @ A_67 @ B_49 ) @ X_50 ) ) ) )).

thf(fact_434_le__supI,axiom,(
    ! [B_49: hoare_1167836817_state > $o,A_67: hoare_1167836817_state > $o,X_50: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_67 @ X_50 )
     => ( ( ord_le827224136tate_o @ B_49 @ X_50 )
       => ( ord_le827224136tate_o @ ( semila1172322802tate_o @ A_67 @ B_49 ) @ X_50 ) ) ) )).

thf(fact_435_sup__absorb1,axiom,(
    ! [Y_30: pname > $o,X_49: pname > $o] :
      ( ( ord_less_eq_pname_o @ Y_30 @ X_49 )
     => ( ( semila1780557381name_o @ X_49 @ Y_30 )
        = X_49 ) ) )).

thf(fact_436_sup__absorb1,axiom,(
    ! [Y_30: hoare_1167836817_state > $o,X_49: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ Y_30 @ X_49 )
     => ( ( semila1172322802tate_o @ X_49 @ Y_30 )
        = X_49 ) ) )).

thf(fact_437_sup__absorb2,axiom,(
    ! [X_48: pname > $o,Y_29: pname > $o] :
      ( ( ord_less_eq_pname_o @ X_48 @ Y_29 )
     => ( ( semila1780557381name_o @ X_48 @ Y_29 )
        = Y_29 ) ) )).

thf(fact_438_sup__absorb2,axiom,(
    ! [X_48: hoare_1167836817_state > $o,Y_29: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ X_48 @ Y_29 )
     => ( ( semila1172322802tate_o @ X_48 @ Y_29 )
        = Y_29 ) ) )).

thf(fact_439_le__supI2,axiom,(
    ! [A_66: pname > $o,X_47: pname > $o,B_48: pname > $o] :
      ( ( ord_less_eq_pname_o @ X_47 @ B_48 )
     => ( ord_less_eq_pname_o @ X_47 @ ( semila1780557381name_o @ A_66 @ B_48 ) ) ) )).

thf(fact_440_le__supI2,axiom,(
    ! [A_66: hoare_1167836817_state > $o,X_47: hoare_1167836817_state > $o,B_48: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ X_47 @ B_48 )
     => ( ord_le827224136tate_o @ X_47 @ ( semila1172322802tate_o @ A_66 @ B_48 ) ) ) )).

thf(fact_441_le__supI1,axiom,(
    ! [B_47: pname > $o,X_46: pname > $o,A_65: pname > $o] :
      ( ( ord_less_eq_pname_o @ X_46 @ A_65 )
     => ( ord_less_eq_pname_o @ X_46 @ ( semila1780557381name_o @ A_65 @ B_47 ) ) ) )).

thf(fact_442_le__supI1,axiom,(
    ! [B_47: hoare_1167836817_state > $o,X_46: hoare_1167836817_state > $o,A_65: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ X_46 @ A_65 )
     => ( ord_le827224136tate_o @ X_46 @ ( semila1172322802tate_o @ A_65 @ B_47 ) ) ) )).

thf(fact_443_le__sup__iff,axiom,(
    ! [X_45: pname > $o,Y_28: pname > $o,Z_12: pname > $o] :
      ( ( ord_less_eq_pname_o @ ( semila1780557381name_o @ X_45 @ Y_28 ) @ Z_12 )
    <=> ( ( ord_less_eq_pname_o @ X_45 @ Z_12 )
        & ( ord_less_eq_pname_o @ Y_28 @ Z_12 ) ) ) )).

thf(fact_444_le__sup__iff,axiom,(
    ! [X_45: hoare_1167836817_state > $o,Y_28: hoare_1167836817_state > $o,Z_12: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ ( semila1172322802tate_o @ X_45 @ Y_28 ) @ Z_12 )
    <=> ( ( ord_le827224136tate_o @ X_45 @ Z_12 )
        & ( ord_le827224136tate_o @ Y_28 @ Z_12 ) ) ) )).

thf(fact_445_le__iff__sup,axiom,(
    ! [X_44: pname > $o,Y_27: pname > $o] :
      ( ( ord_less_eq_pname_o @ X_44 @ Y_27 )
    <=> ( ( semila1780557381name_o @ X_44 @ Y_27 )
        = Y_27 ) ) )).

thf(fact_446_le__iff__sup,axiom,(
    ! [X_44: hoare_1167836817_state > $o,Y_27: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ X_44 @ Y_27 )
    <=> ( ( semila1172322802tate_o @ X_44 @ Y_27 )
        = Y_27 ) ) )).

thf(fact_447_sup__ge2,axiom,(
    ! [Y_26: pname > $o,X_43: pname > $o] :
      ( ord_less_eq_pname_o @ Y_26 @ ( semila1780557381name_o @ X_43 @ Y_26 ) ) )).

thf(fact_448_sup__ge2,axiom,(
    ! [Y_26: hoare_1167836817_state > $o,X_43: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ Y_26 @ ( semila1172322802tate_o @ X_43 @ Y_26 ) ) )).

thf(fact_449_inf__sup__ord_I4_J,axiom,(
    ! [Y_25: pname > $o,X_42: pname > $o] :
      ( ord_less_eq_pname_o @ Y_25 @ ( semila1780557381name_o @ X_42 @ Y_25 ) ) )).

thf(fact_450_inf__sup__ord_I4_J,axiom,(
    ! [Y_25: hoare_1167836817_state > $o,X_42: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ Y_25 @ ( semila1172322802tate_o @ X_42 @ Y_25 ) ) )).

thf(fact_451_sup__ge1,axiom,(
    ! [X_41: pname > $o,Y_24: pname > $o] :
      ( ord_less_eq_pname_o @ X_41 @ ( semila1780557381name_o @ X_41 @ Y_24 ) ) )).

thf(fact_452_sup__ge1,axiom,(
    ! [X_41: hoare_1167836817_state > $o,Y_24: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ X_41 @ ( semila1172322802tate_o @ X_41 @ Y_24 ) ) )).

thf(fact_453_inf__sup__ord_I3_J,axiom,(
    ! [X_40: pname > $o,Y_23: pname > $o] :
      ( ord_less_eq_pname_o @ X_40 @ ( semila1780557381name_o @ X_40 @ Y_23 ) ) )).

thf(fact_454_inf__sup__ord_I3_J,axiom,(
    ! [X_40: hoare_1167836817_state > $o,Y_23: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ X_40 @ ( semila1172322802tate_o @ X_40 @ Y_23 ) ) )).

thf(fact_455_sup__eq__bot__iff,axiom,(
    ! [X_39: pname > $o,Y_22: pname > $o] :
      ( ( ( semila1780557381name_o @ X_39 @ Y_22 )
        = bot_bot_pname_o )
    <=> ( ( X_39 = bot_bot_pname_o )
        & ( Y_22 = bot_bot_pname_o ) ) ) )).

thf(fact_456_sup__eq__bot__iff,axiom,(
    ! [X_39: hoare_1167836817_state > $o,Y_22: hoare_1167836817_state > $o] :
      ( ( ( semila1172322802tate_o @ X_39 @ Y_22 )
        = bot_bo70021908tate_o )
    <=> ( ( X_39 = bot_bo70021908tate_o )
        & ( Y_22 = bot_bo70021908tate_o ) ) ) )).

thf(fact_457_sup__bot__right,axiom,(
    ! [X_38: pname > $o] :
      ( ( semila1780557381name_o @ X_38 @ bot_bot_pname_o )
      = X_38 ) )).

thf(fact_458_sup__bot__right,axiom,(
    ! [X_38: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ X_38 @ bot_bo70021908tate_o )
      = X_38 ) )).

thf(fact_459_sup__bot__left,axiom,(
    ! [X_37: pname > $o] :
      ( ( semila1780557381name_o @ bot_bot_pname_o @ X_37 )
      = X_37 ) )).

thf(fact_460_sup__bot__left,axiom,(
    ! [X_37: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ bot_bo70021908tate_o @ X_37 )
      = X_37 ) )).

thf(fact_461_folding__one__idem_Ounion__idem,axiom,(
    ! [B_46: ( pname > $o ) > $o,A_64: ( pname > $o ) > $o,F_32: ( pname > $o ) > ( pname > $o ) > pname > $o,F_31: ( ( pname > $o ) > $o ) > pname > $o] :
      ( ( finite697516351name_o @ F_32 @ F_31 )
     => ( ( finite297249702name_o @ A_64 )
       => ( ( A_64 != bot_bot_pname_o_o )
         => ( ( finite297249702name_o @ B_46 )
           => ( ( B_46 != bot_bot_pname_o_o )
             => ( ( F_31 @ ( semila181081674me_o_o @ A_64 @ B_46 ) )
                = ( F_32 @ ( F_31 @ A_64 ) @ ( F_31 @ B_46 ) ) ) ) ) ) ) ) )).

thf(fact_462_folding__one__idem_Ounion__idem,axiom,(
    ! [B_46: ( hoare_1167836817_state > $o ) > $o,A_64: ( hoare_1167836817_state > $o ) > $o,F_32: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,F_31: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o] :
      ( ( finite671847800tate_o @ F_32 @ F_31 )
     => ( ( finite1380128977tate_o @ A_64 )
       => ( ( A_64 != bot_bo691907561te_o_o )
         => ( ( finite1380128977tate_o @ B_46 )
           => ( ( B_46 != bot_bo691907561te_o_o )
             => ( ( F_31 @ ( semila866907787te_o_o @ A_64 @ B_46 ) )
                = ( F_32 @ ( F_31 @ A_64 ) @ ( F_31 @ B_46 ) ) ) ) ) ) ) ) )).

thf(fact_463_folding__one__idem_Ounion__idem,axiom,(
    ! [B_46: pname > $o,A_64: pname > $o,F_32: pname > pname > pname,F_31: ( pname > $o ) > pname] :
      ( ( finite89670078_pname @ F_32 @ F_31 )
     => ( ( finite_finite_pname @ A_64 )
       => ( ( A_64 != bot_bot_pname_o )
         => ( ( finite_finite_pname @ B_46 )
           => ( ( B_46 != bot_bot_pname_o )
             => ( ( F_31 @ ( semila1780557381name_o @ A_64 @ B_46 ) )
                = ( F_32 @ ( F_31 @ A_64 ) @ ( F_31 @ B_46 ) ) ) ) ) ) ) ) )).

thf(fact_464_folding__one__idem_Ounion__idem,axiom,(
    ! [B_46: hoare_1167836817_state > $o,A_64: hoare_1167836817_state > $o,F_32: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_31: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
      ( ( finite806517911_state @ F_32 @ F_31 )
     => ( ( finite1084549118_state @ A_64 )
       => ( ( A_64 != bot_bo70021908tate_o )
         => ( ( finite1084549118_state @ B_46 )
           => ( ( B_46 != bot_bo70021908tate_o )
             => ( ( F_31 @ ( semila1172322802tate_o @ A_64 @ B_46 ) )
                = ( F_32 @ ( F_31 @ A_64 ) @ ( F_31 @ B_46 ) ) ) ) ) ) ) ) )).

thf(fact_465_folding__one__idem_Osubset__idem,axiom,(
    ! [B_45: ( pname > $o ) > $o,A_63: ( pname > $o ) > $o,F_30: ( pname > $o ) > ( pname > $o ) > pname > $o,F_29: ( ( pname > $o ) > $o ) > pname > $o] :
      ( ( finite697516351name_o @ F_30 @ F_29 )
     => ( ( finite297249702name_o @ A_63 )
       => ( ( B_45 != bot_bot_pname_o_o )
         => ( ( ord_le1205211808me_o_o @ B_45 @ A_63 )
           => ( ( F_30 @ ( F_29 @ B_45 ) @ ( F_29 @ A_63 ) )
              = ( F_29 @ A_63 ) ) ) ) ) ) )).

thf(fact_466_folding__one__idem_Osubset__idem,axiom,(
    ! [B_45: ( hoare_1167836817_state > $o ) > $o,A_63: ( hoare_1167836817_state > $o ) > $o,F_30: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,F_29: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o] :
      ( ( finite671847800tate_o @ F_30 @ F_29 )
     => ( ( finite1380128977tate_o @ A_63 )
       => ( ( B_45 != bot_bo691907561te_o_o )
         => ( ( ord_le741939125te_o_o @ B_45 @ A_63 )
           => ( ( F_30 @ ( F_29 @ B_45 ) @ ( F_29 @ A_63 ) )
              = ( F_29 @ A_63 ) ) ) ) ) ) )).

thf(fact_467_folding__one__idem_Osubset__idem,axiom,(
    ! [B_45: pname > $o,A_63: pname > $o,F_30: pname > pname > pname,F_29: ( pname > $o ) > pname] :
      ( ( finite89670078_pname @ F_30 @ F_29 )
     => ( ( finite_finite_pname @ A_63 )
       => ( ( B_45 != bot_bot_pname_o )
         => ( ( ord_less_eq_pname_o @ B_45 @ A_63 )
           => ( ( F_30 @ ( F_29 @ B_45 ) @ ( F_29 @ A_63 ) )
              = ( F_29 @ A_63 ) ) ) ) ) ) )).

thf(fact_468_folding__one__idem_Osubset__idem,axiom,(
    ! [B_45: hoare_1167836817_state > $o,A_63: hoare_1167836817_state > $o,F_30: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_29: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
      ( ( finite806517911_state @ F_30 @ F_29 )
     => ( ( finite1084549118_state @ A_63 )
       => ( ( B_45 != bot_bo70021908tate_o )
         => ( ( ord_le827224136tate_o @ B_45 @ A_63 )
           => ( ( F_30 @ ( F_29 @ B_45 ) @ ( F_29 @ A_63 ) )
              = ( F_29 @ A_63 ) ) ) ) ) ) )).

thf(fact_469_hoare__derivs_OSkip,axiom,(
    ! [G_7: hoare_1167836817_state > $o,P_11: state > state > $o] :
      ( hoare_123228589_state @ G_7 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_11 @ skip @ P_11 ) @ bot_bo70021908tate_o ) ) )).

thf(fact_470_folding__one__idem_Oinsert__idem,axiom,(
    ! [X_36: pname,A_62: pname > $o,F_28: pname > pname > pname,F_27: ( pname > $o ) > pname] :
      ( ( finite89670078_pname @ F_28 @ F_27 )
     => ( ( finite_finite_pname @ A_62 )
       => ( ( A_62 != bot_bot_pname_o )
         => ( ( F_27 @ ( insert_pname @ X_36 @ A_62 ) )
            = ( F_28 @ X_36 @ ( F_27 @ A_62 ) ) ) ) ) ) )).

thf(fact_471_folding__one__idem_Oinsert__idem,axiom,(
    ! [X_36: hoare_1167836817_state,A_62: hoare_1167836817_state > $o,F_28: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_27: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
      ( ( finite806517911_state @ F_28 @ F_27 )
     => ( ( finite1084549118_state @ A_62 )
       => ( ( A_62 != bot_bo70021908tate_o )
         => ( ( F_27 @ ( insert2134838167_state @ X_36 @ A_62 ) )
            = ( F_28 @ X_36 @ ( F_27 @ A_62 ) ) ) ) ) ) )).

thf(fact_472_folding__one__idem_Oinsert__idem,axiom,(
    ! [X_36: pname > $o,A_62: ( pname > $o ) > $o,F_28: ( pname > $o ) > ( pname > $o ) > pname > $o,F_27: ( ( pname > $o ) > $o ) > pname > $o] :
      ( ( finite697516351name_o @ F_28 @ F_27 )
     => ( ( finite297249702name_o @ A_62 )
       => ( ( A_62 != bot_bot_pname_o_o )
         => ( ( F_27 @ ( insert_pname_o @ X_36 @ A_62 ) )
            = ( F_28 @ X_36 @ ( F_27 @ A_62 ) ) ) ) ) ) )).

thf(fact_473_folding__one__idem_Oinsert__idem,axiom,(
    ! [X_36: hoare_1167836817_state > $o,A_62: ( hoare_1167836817_state > $o ) > $o,F_28: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,F_27: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o] :
      ( ( finite671847800tate_o @ F_28 @ F_27 )
     => ( ( finite1380128977tate_o @ A_62 )
       => ( ( A_62 != bot_bo691907561te_o_o )
         => ( ( F_27 @ ( insert999278200tate_o @ X_36 @ A_62 ) )
            = ( F_28 @ X_36 @ ( F_27 @ A_62 ) ) ) ) ) ) )).

thf(fact_474_finite__ne__induct,axiom,(
    ! [P_10: ( pname > $o ) > $o,F_25: pname > $o] :
      ( ( finite_finite_pname @ F_25 )
     => ( ( F_25 != bot_bot_pname_o )
       => ( ! [X_5: pname] :
              ( P_10 @ ( insert_pname @ X_5 @ bot_bot_pname_o ) )
         => ( ! [X_5: pname,F_26: pname > $o] :
                ( ( finite_finite_pname @ F_26 )
               => ( ( F_26 != bot_bot_pname_o )
                 => ( ~ ( member_pname @ X_5 @ F_26 )
                   => ( ( P_10 @ F_26 )
                     => ( P_10 @ ( insert_pname @ X_5 @ F_26 ) ) ) ) ) )
           => ( P_10 @ F_25 ) ) ) ) ) )).

thf(fact_475_finite__ne__induct,axiom,(
    ! [P_10: ( hoare_1167836817_state > $o ) > $o,F_25: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ F_25 )
     => ( ( F_25 != bot_bo70021908tate_o )
       => ( ! [X_5: hoare_1167836817_state] :
              ( P_10 @ ( insert2134838167_state @ X_5 @ bot_bo70021908tate_o ) )
         => ( ! [X_5: hoare_1167836817_state,F_26: hoare_1167836817_state > $o] :
                ( ( finite1084549118_state @ F_26 )
               => ( ( F_26 != bot_bo70021908tate_o )
                 => ( ~ ( member2058392318_state @ X_5 @ F_26 )
                   => ( ( P_10 @ F_26 )
                     => ( P_10 @ ( insert2134838167_state @ X_5 @ F_26 ) ) ) ) ) )
           => ( P_10 @ F_25 ) ) ) ) ) )).

thf(fact_476_finite__ne__induct,axiom,(
    ! [P_10: ( ( pname > $o ) > $o ) > $o,F_25: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ F_25 )
     => ( ( F_25 != bot_bot_pname_o_o )
       => ( ! [X_5: pname > $o] :
              ( P_10 @ ( insert_pname_o @ X_5 @ bot_bot_pname_o_o ) )
         => ( ! [X_5: pname > $o,F_26: ( pname > $o ) > $o] :
                ( ( finite297249702name_o @ F_26 )
               => ( ( F_26 != bot_bot_pname_o_o )
                 => ( ~ ( member_pname_o @ X_5 @ F_26 )
                   => ( ( P_10 @ F_26 )
                     => ( P_10 @ ( insert_pname_o @ X_5 @ F_26 ) ) ) ) ) )
           => ( P_10 @ F_25 ) ) ) ) ) )).

thf(fact_477_finite__ne__induct,axiom,(
    ! [P_10: ( ( hoare_1167836817_state > $o ) > $o ) > $o,F_25: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ F_25 )
     => ( ( F_25 != bot_bo691907561te_o_o )
       => ( ! [X_5: hoare_1167836817_state > $o] :
              ( P_10 @ ( insert999278200tate_o @ X_5 @ bot_bo691907561te_o_o ) )
         => ( ! [X_5: hoare_1167836817_state > $o,F_26: ( hoare_1167836817_state > $o ) > $o] :
                ( ( finite1380128977tate_o @ F_26 )
               => ( ( F_26 != bot_bo691907561te_o_o )
                 => ( ~ ( member864234961tate_o @ X_5 @ F_26 )
                   => ( ( P_10 @ F_26 )
                     => ( P_10 @ ( insert999278200tate_o @ X_5 @ F_26 ) ) ) ) ) )
           => ( P_10 @ F_25 ) ) ) ) ) )).

thf(fact_478_com_Osimps_I19_J,axiom,(
    ! [Pname: pname] :
      ( ( body_1 @ Pname )
     != skip ) )).

thf(fact_479_com_Osimps_I18_J,axiom,(
    ! [Pname: pname] :
      ( skip
     != ( body_1 @ Pname ) ) )).

thf(fact_480_WT_OSkip,axiom,
    ( wt @ skip )).

thf(fact_481_folding__one__idem_Oin__idem,axiom,(
    ! [X_35: pname,A_61: pname > $o,F_24: pname > pname > pname,F_23: ( pname > $o ) > pname] :
      ( ( finite89670078_pname @ F_24 @ F_23 )
     => ( ( finite_finite_pname @ A_61 )
       => ( ( member_pname @ X_35 @ A_61 )
         => ( ( F_24 @ X_35 @ ( F_23 @ A_61 ) )
            = ( F_23 @ A_61 ) ) ) ) ) )).

thf(fact_482_folding__one__idem_Oin__idem,axiom,(
    ! [X_35: hoare_1167836817_state,A_61: hoare_1167836817_state > $o,F_24: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_23: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
      ( ( finite806517911_state @ F_24 @ F_23 )
     => ( ( finite1084549118_state @ A_61 )
       => ( ( member2058392318_state @ X_35 @ A_61 )
         => ( ( F_24 @ X_35 @ ( F_23 @ A_61 ) )
            = ( F_23 @ A_61 ) ) ) ) ) )).

thf(fact_483_folding__one__idem_Oin__idem,axiom,(
    ! [X_35: pname > $o,A_61: ( pname > $o ) > $o,F_24: ( pname > $o ) > ( pname > $o ) > pname > $o,F_23: ( ( pname > $o ) > $o ) > pname > $o] :
      ( ( finite697516351name_o @ F_24 @ F_23 )
     => ( ( finite297249702name_o @ A_61 )
       => ( ( member_pname_o @ X_35 @ A_61 )
         => ( ( F_24 @ X_35 @ ( F_23 @ A_61 ) )
            = ( F_23 @ A_61 ) ) ) ) ) )).

thf(fact_484_folding__one__idem_Oin__idem,axiom,(
    ! [X_35: hoare_1167836817_state > $o,A_61: ( hoare_1167836817_state > $o ) > $o,F_24: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,F_23: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o] :
      ( ( finite671847800tate_o @ F_24 @ F_23 )
     => ( ( finite1380128977tate_o @ A_61 )
       => ( ( member864234961tate_o @ X_35 @ A_61 )
         => ( ( F_24 @ X_35 @ ( F_23 @ A_61 ) )
            = ( F_23 @ A_61 ) ) ) ) ) )).

thf(fact_485_folding__one__idem_Ohom__commute,axiom,(
    ! [N_1: ( pname > $o ) > $o,H: ( pname > $o ) > pname > $o,F_22: ( pname > $o ) > ( pname > $o ) > pname > $o,F_21: ( ( pname > $o ) > $o ) > pname > $o] :
      ( ( finite697516351name_o @ F_22 @ F_21 )
     => ( ! [X_5: pname > $o,Y_2: pname > $o] :
            ( ( H @ ( F_22 @ X_5 @ Y_2 ) )
            = ( F_22 @ ( H @ X_5 ) @ ( H @ Y_2 ) ) )
       => ( ( finite297249702name_o @ N_1 )
         => ( ( N_1 != bot_bot_pname_o_o )
           => ( ( H @ ( F_21 @ N_1 ) )
              = ( F_21 @ ( image_1085733413name_o @ H @ N_1 ) ) ) ) ) ) ) )).

thf(fact_486_folding__one__idem_Ohom__commute,axiom,(
    ! [N_1: ( hoare_1167836817_state > $o ) > $o,H: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,F_22: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,F_21: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o] :
      ( ( finite671847800tate_o @ F_22 @ F_21 )
     => ( ! [X_5: hoare_1167836817_state > $o,Y_2: hoare_1167836817_state > $o] :
            ( ( H @ ( F_22 @ X_5 @ Y_2 ) )
            = ( F_22 @ ( H @ X_5 ) @ ( H @ Y_2 ) ) )
       => ( ( finite1380128977tate_o @ N_1 )
         => ( ( N_1 != bot_bo691907561te_o_o )
           => ( ( H @ ( F_21 @ N_1 ) )
              = ( F_21 @ ( image_1488525317tate_o @ H @ N_1 ) ) ) ) ) ) ) )).

thf(fact_487_folding__one__idem_Ohom__commute,axiom,(
    ! [N_1: pname > $o,H: pname > pname,F_22: pname > pname > pname,F_21: ( pname > $o ) > pname] :
      ( ( finite89670078_pname @ F_22 @ F_21 )
     => ( ! [X_5: pname,Y_2: pname] :
            ( ( H @ ( F_22 @ X_5 @ Y_2 ) )
            = ( F_22 @ ( H @ X_5 ) @ ( H @ Y_2 ) ) )
       => ( ( finite_finite_pname @ N_1 )
         => ( ( N_1 != bot_bot_pname_o )
           => ( ( H @ ( F_21 @ N_1 ) )
              = ( F_21 @ ( image_pname_pname @ H @ N_1 ) ) ) ) ) ) ) )).

thf(fact_488_folding__one__idem_Ohom__commute,axiom,(
    ! [N_1: hoare_1167836817_state > $o,H: hoare_1167836817_state > hoare_1167836817_state,F_22: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_21: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
      ( ( finite806517911_state @ F_22 @ F_21 )
     => ( ! [X_5: hoare_1167836817_state,Y_2: hoare_1167836817_state] :
            ( ( H @ ( F_22 @ X_5 @ Y_2 ) )
            = ( F_22 @ ( H @ X_5 ) @ ( H @ Y_2 ) ) )
       => ( ( finite1084549118_state @ N_1 )
         => ( ( N_1 != bot_bo70021908tate_o )
           => ( ( H @ ( F_21 @ N_1 ) )
              = ( F_21 @ ( image_31595733_state @ H @ N_1 ) ) ) ) ) ) ) )).

thf(fact_489_LoopF,axiom,(
    ! [G_6: hoare_1167836817_state > $o,P_9: state > state > $o,B_44: state > $o,C_23: com] :
      ( hoare_123228589_state @ G_6
      @ ( insert2134838167_state
        @ ( hoare_908217195_state
          @ ^ [Z_11: state,S_3: state] :
              ( (&) @ ( P_9 @ Z_11 @ S_3 ) @ ( (~) @ ( B_44 @ S_3 ) ) )
          @ ( while @ B_44 @ C_23 )
          @ P_9 )
        @ bot_bo70021908tate_o ) ) )).

thf(fact_490_Comp,axiom,(
    ! [D_2: com,R_1: state > state > $o,G_5: hoare_1167836817_state > $o,P_8: state > state > $o,C_22: com,Q_4: state > state > $o] :
      ( ( hoare_123228589_state @ G_5 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_8 @ C_22 @ Q_4 ) @ bot_bo70021908tate_o ) )
     => ( ( hoare_123228589_state @ G_5 @ ( insert2134838167_state @ ( hoare_908217195_state @ Q_4 @ D_2 @ R_1 ) @ bot_bo70021908tate_o ) )
       => ( hoare_123228589_state @ G_5 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_8 @ ( semi @ C_22 @ D_2 ) @ R_1 ) @ bot_bo70021908tate_o ) ) ) ) )).

thf(fact_491_the__elem__def,axiom,(
    ! [X_34: pname > $o] :
      ( ( the_elem_pname @ X_34 )
      = ( the_pname
        @ ^ [X_5: pname] :
            ( X_34
            = ( insert_pname @ X_5 @ bot_bot_pname_o ) ) ) ) )).

thf(fact_492_the__elem__def,axiom,(
    ! [X_34: hoare_1167836817_state > $o] :
      ( ( the_el323660082_state @ X_34 )
      = ( the_Ho310147232_state
        @ ^ [X_5: hoare_1167836817_state] :
            ( X_34
            = ( insert2134838167_state @ X_5 @ bot_bo70021908tate_o ) ) ) ) )).

thf(fact_493_WTs__elim__cases_I6_J,axiom,(
    ! [B_42: state > $o,C_21: com] :
      ( ( wt @ ( while @ B_42 @ C_21 ) )
     => ( wt @ C_21 ) ) )).

thf(fact_494_WTs__elim__cases_I4_J,axiom,(
    ! [C1: com,C2: com] :
      ( ( wt @ ( semi @ C1 @ C2 ) )
     => ~ ( ( wt @ C1 )
         => ~ ( wt @ C2 ) ) ) )).

thf(fact_495_com_Osimps_I46_J,axiom,(
    ! [Com1_1: com,Com2_1: com,Fun: state > $o,Com_1: com] :
      ( ( semi @ Com1_1 @ Com2_1 )
     != ( while @ Fun @ Com_1 ) ) )).

thf(fact_496_com_Osimps_I47_J,axiom,(
    ! [Fun: state > $o,Com_1: com,Com1_1: com,Com2_1: com] :
      ( ( while @ Fun @ Com_1 )
     != ( semi @ Com1_1 @ Com2_1 ) ) )).

thf(fact_497_com_Osimps_I3_J,axiom,(
    ! [Com1_1: com,Com2_1: com,Com1: com,Com2: com] :
      ( ( ( semi @ Com1_1 @ Com2_1 )
        = ( semi @ Com1 @ Com2 ) )
    <=> ( ( Com1_1 = Com1 )
        & ( Com2_1 = Com2 ) ) ) )).

thf(fact_498_com_Osimps_I5_J,axiom,(
    ! [Fun_1: state > $o,Com_2: com,Fun: state > $o,Com_1: com] :
      ( ( ( while @ Fun_1 @ Com_2 )
        = ( while @ Fun @ Com_1 ) )
    <=> ( ( Fun_1 = Fun )
        & ( Com_2 = Com_1 ) ) ) )).

thf(fact_499_com_Osimps_I59_J,axiom,(
    ! [Pname: pname,Fun_1: state > $o,Com_2: com] :
      ( ( body_1 @ Pname )
     != ( while @ Fun_1 @ Com_2 ) ) )).

thf(fact_500_com_Osimps_I58_J,axiom,(
    ! [Fun_1: state > $o,Com_2: com,Pname: pname] :
      ( ( while @ Fun_1 @ Com_2 )
     != ( body_1 @ Pname ) ) )).

thf(fact_501_While,axiom,(
    ! [B_42: state > $o,C_21: com] :
      ( ( wt @ C_21 )
     => ( wt @ ( while @ B_42 @ C_21 ) ) ) )).

thf(fact_502_com_Osimps_I16_J,axiom,(
    ! [Fun: state > $o,Com_1: com] :
      ( skip
     != ( while @ Fun @ Com_1 ) ) )).

thf(fact_503_com_Osimps_I17_J,axiom,(
    ! [Fun: state > $o,Com_1: com] :
      ( ( while @ Fun @ Com_1 )
     != skip ) )).

thf(fact_504_com_Osimps_I49_J,axiom,(
    ! [Pname: pname,Com1_1: com,Com2_1: com] :
      ( ( body_1 @ Pname )
     != ( semi @ Com1_1 @ Com2_1 ) ) )).

thf(fact_505_com_Osimps_I48_J,axiom,(
    ! [Com1_1: com,Com2_1: com,Pname: pname] :
      ( ( semi @ Com1_1 @ Com2_1 )
     != ( body_1 @ Pname ) ) )).

thf(fact_506_WT_OSemi,axiom,(
    ! [C1: com,C0: com] :
      ( ( wt @ C0 )
     => ( ( wt @ C1 )
       => ( wt @ ( semi @ C0 @ C1 ) ) ) ) )).

thf(fact_507_com_Osimps_I12_J,axiom,(
    ! [Com1: com,Com2: com] :
      ( skip
     != ( semi @ Com1 @ Com2 ) ) )).

thf(fact_508_com_Osimps_I13_J,axiom,(
    ! [Com1: com,Com2: com] :
      ( ( semi @ Com1 @ Com2 )
     != skip ) )).

thf(fact_509_folding__one_Oinsert,axiom,(
    ! [X_33: pname,A_60: pname > $o,F_20: pname > pname > pname,F_19: ( pname > $o ) > pname] :
      ( ( finite1282449217_pname @ F_20 @ F_19 )
     => ( ( finite_finite_pname @ A_60 )
       => ( ~ ( member_pname @ X_33 @ A_60 )
         => ( ( A_60 != bot_bot_pname_o )
           => ( ( F_19 @ ( insert_pname @ X_33 @ A_60 ) )
              = ( F_20 @ X_33 @ ( F_19 @ A_60 ) ) ) ) ) ) ) )).

thf(fact_510_folding__one_Oinsert,axiom,(
    ! [X_33: hoare_1167836817_state,A_60: hoare_1167836817_state > $o,F_20: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_19: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
      ( ( finite1074406356_state @ F_20 @ F_19 )
     => ( ( finite1084549118_state @ A_60 )
       => ( ~ ( member2058392318_state @ X_33 @ A_60 )
         => ( ( A_60 != bot_bo70021908tate_o )
           => ( ( F_19 @ ( insert2134838167_state @ X_33 @ A_60 ) )
              = ( F_20 @ X_33 @ ( F_19 @ A_60 ) ) ) ) ) ) ) )).

thf(fact_511_folding__one_Oinsert,axiom,(
    ! [X_33: pname > $o,A_60: ( pname > $o ) > $o,F_20: ( pname > $o ) > ( pname > $o ) > pname > $o,F_19: ( ( pname > $o ) > $o ) > pname > $o] :
      ( ( finite349908348name_o @ F_20 @ F_19 )
     => ( ( finite297249702name_o @ A_60 )
       => ( ~ ( member_pname_o @ X_33 @ A_60 )
         => ( ( A_60 != bot_bot_pname_o_o )
           => ( ( F_19 @ ( insert_pname_o @ X_33 @ A_60 ) )
              = ( F_20 @ X_33 @ ( F_19 @ A_60 ) ) ) ) ) ) ) )).

thf(fact_512_folding__one_Oinsert,axiom,(
    ! [X_33: hoare_1167836817_state > $o,A_60: ( hoare_1167836817_state > $o ) > $o,F_20: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,F_19: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o] :
      ( ( finite979047547tate_o @ F_20 @ F_19 )
     => ( ( finite1380128977tate_o @ A_60 )
       => ( ~ ( member864234961tate_o @ X_33 @ A_60 )
         => ( ( A_60 != bot_bo691907561te_o_o )
           => ( ( F_19 @ ( insert999278200tate_o @ X_33 @ A_60 ) )
              = ( F_20 @ X_33 @ ( F_19 @ A_60 ) ) ) ) ) ) ) )).

thf(fact_513_folding__one_Osingleton,axiom,(
    ! [X_32: pname,F_18: pname > pname > pname,F_17: ( pname > $o ) > pname] :
      ( ( finite1282449217_pname @ F_18 @ F_17 )
     => ( ( F_17 @ ( insert_pname @ X_32 @ bot_bot_pname_o ) )
        = X_32 ) ) )).

thf(fact_514_folding__one_Osingleton,axiom,(
    ! [X_32: hoare_1167836817_state,F_18: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_17: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
      ( ( finite1074406356_state @ F_18 @ F_17 )
     => ( ( F_17 @ ( insert2134838167_state @ X_32 @ bot_bo70021908tate_o ) )
        = X_32 ) ) )).

thf(fact_515_folding__one_Oclosed,axiom,(
    ! [A_59: pname > $o,F_16: pname > pname > pname,F_15: ( pname > $o ) > pname] :
      ( ( finite1282449217_pname @ F_16 @ F_15 )
     => ( ( finite_finite_pname @ A_59 )
       => ( ( A_59 != bot_bot_pname_o )
         => ( ! [X_5: pname,Y_2: pname] :
                ( member_pname @ ( F_16 @ X_5 @ Y_2 ) @ ( insert_pname @ X_5 @ ( insert_pname @ Y_2 @ bot_bot_pname_o ) ) )
           => ( member_pname @ ( F_15 @ A_59 ) @ A_59 ) ) ) ) ) )).

thf(fact_516_folding__one_Oclosed,axiom,(
    ! [A_59: hoare_1167836817_state > $o,F_16: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_15: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
      ( ( finite1074406356_state @ F_16 @ F_15 )
     => ( ( finite1084549118_state @ A_59 )
       => ( ( A_59 != bot_bo70021908tate_o )
         => ( ! [X_5: hoare_1167836817_state,Y_2: hoare_1167836817_state] :
                ( member2058392318_state @ ( F_16 @ X_5 @ Y_2 ) @ ( insert2134838167_state @ X_5 @ ( insert2134838167_state @ Y_2 @ bot_bo70021908tate_o ) ) )
           => ( member2058392318_state @ ( F_15 @ A_59 ) @ A_59 ) ) ) ) ) )).

thf(fact_517_folding__one_Oclosed,axiom,(
    ! [A_59: ( pname > $o ) > $o,F_16: ( pname > $o ) > ( pname > $o ) > pname > $o,F_15: ( ( pname > $o ) > $o ) > pname > $o] :
      ( ( finite349908348name_o @ F_16 @ F_15 )
     => ( ( finite297249702name_o @ A_59 )
       => ( ( A_59 != bot_bot_pname_o_o )
         => ( ! [X_5: pname > $o,Y_2: pname > $o] :
                ( member_pname_o @ ( F_16 @ X_5 @ Y_2 ) @ ( insert_pname_o @ X_5 @ ( insert_pname_o @ Y_2 @ bot_bot_pname_o_o ) ) )
           => ( member_pname_o @ ( F_15 @ A_59 ) @ A_59 ) ) ) ) ) )).

thf(fact_518_folding__one_Oclosed,axiom,(
    ! [A_59: ( hoare_1167836817_state > $o ) > $o,F_16: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,F_15: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o] :
      ( ( finite979047547tate_o @ F_16 @ F_15 )
     => ( ( finite1380128977tate_o @ A_59 )
       => ( ( A_59 != bot_bo691907561te_o_o )
         => ( ! [X_5: hoare_1167836817_state > $o,Y_2: hoare_1167836817_state > $o] :
                ( member864234961tate_o @ ( F_16 @ X_5 @ Y_2 ) @ ( insert999278200tate_o @ X_5 @ ( insert999278200tate_o @ Y_2 @ bot_bo691907561te_o_o ) ) )
           => ( member864234961tate_o @ ( F_15 @ A_59 ) @ A_59 ) ) ) ) ) )).

thf(fact_519_triple_Oexhaust,axiom,(
    ! [Y_21: hoare_1167836817_state] :
      ~ ! [Fun1: state > state > $o,Com: com,Fun2: state > state > $o] :
          ( Y_21
         != ( hoare_908217195_state @ Fun1 @ Com @ Fun2 ) ) )).

thf(fact_520_image__cong,axiom,(
    ! [F_14: pname > hoare_1167836817_state,G_4: pname > hoare_1167836817_state,M: pname > $o,N: pname > $o] :
      ( ( M = N )
     => ( ! [X_5: pname] :
            ( ( member_pname @ X_5 @ N )
           => ( ( F_14 @ X_5 )
              = ( G_4 @ X_5 ) ) )
       => ( ( image_575578384_state @ F_14 @ M )
          = ( image_575578384_state @ G_4 @ N ) ) ) ) )).

thf(fact_521_Collect__mono,axiom,(
    ! [Q_3: hoare_1167836817_state > $o,P_7: hoare_1167836817_state > $o] :
      ( ! [X_5: hoare_1167836817_state] :
          ( ( P_7 @ X_5 )
         => ( Q_3 @ X_5 ) )
     => ( ord_le827224136tate_o @ ( collec1027672124_state @ P_7 ) @ ( collec1027672124_state @ Q_3 ) ) ) )).

thf(fact_522_Collect__mono,axiom,(
    ! [Q_3: pname > $o,P_7: pname > $o] :
      ( ! [X_5: pname] :
          ( ( P_7 @ X_5 )
         => ( Q_3 @ X_5 ) )
     => ( ord_less_eq_pname_o @ ( collect_pname @ P_7 ) @ ( collect_pname @ Q_3 ) ) ) )).

thf(fact_523_Collect__mono,axiom,(
    ! [Q_3: ( pname > $o ) > $o,P_7: ( pname > $o ) > $o] :
      ( ! [X_5: pname > $o] :
          ( ( P_7 @ X_5 )
         => ( Q_3 @ X_5 ) )
     => ( ord_le1205211808me_o_o @ ( collect_pname_o @ P_7 ) @ ( collect_pname_o @ Q_3 ) ) ) )).

thf(fact_524_Collect__mono,axiom,(
    ! [Q_3: ( hoare_1167836817_state > $o ) > $o,P_7: ( hoare_1167836817_state > $o ) > $o] :
      ( ! [X_5: hoare_1167836817_state > $o] :
          ( ( P_7 @ X_5 )
         => ( Q_3 @ X_5 ) )
     => ( ord_le741939125te_o_o @ ( collec269976083tate_o @ P_7 ) @ ( collec269976083tate_o @ Q_3 ) ) ) )).

thf(fact_525_predicate1I,axiom,(
    ! [Q_2: pname > $o,P_6: pname > $o] :
      ( ! [X_5: pname] :
          ( ( P_6 @ X_5 )
         => ( Q_2 @ X_5 ) )
     => ( ord_less_eq_pname_o @ P_6 @ Q_2 ) ) )).

thf(fact_526_predicate1I,axiom,(
    ! [Q_2: hoare_1167836817_state > $o,P_6: hoare_1167836817_state > $o] :
      ( ! [X_5: hoare_1167836817_state] :
          ( ( P_6 @ X_5 )
         => ( Q_2 @ X_5 ) )
     => ( ord_le827224136tate_o @ P_6 @ Q_2 ) ) )).

thf(fact_527_mk__disjoint__insert,axiom,(
    ! [A_58: pname,A_57: pname > $o] :
      ( ( member_pname @ A_58 @ A_57 )
     => ? [B_43: pname > $o] :
          ( ( A_57
            = ( insert_pname @ A_58 @ B_43 ) )
          & ~ ( member_pname @ A_58 @ B_43 ) ) ) )).

thf(fact_528_mk__disjoint__insert,axiom,(
    ! [A_58: hoare_1167836817_state,A_57: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ A_58 @ A_57 )
     => ? [B_43: hoare_1167836817_state > $o] :
          ( ( A_57
            = ( insert2134838167_state @ A_58 @ B_43 ) )
          & ~ ( member2058392318_state @ A_58 @ B_43 ) ) ) )).

thf(fact_529_Set_Oset__insert,axiom,(
    ! [X_31: pname,A_56: pname > $o] :
      ( ( member_pname @ X_31 @ A_56 )
     => ~ ! [B_43: pname > $o] :
            ( ( A_56
              = ( insert_pname @ X_31 @ B_43 ) )
           => ( member_pname @ X_31 @ B_43 ) ) ) )).

thf(fact_530_Set_Oset__insert,axiom,(
    ! [X_31: hoare_1167836817_state,A_56: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ X_31 @ A_56 )
     => ~ ! [B_43: hoare_1167836817_state > $o] :
            ( ( A_56
              = ( insert2134838167_state @ X_31 @ B_43 ) )
           => ( member2058392318_state @ X_31 @ B_43 ) ) ) )).

thf(fact_531_equals0I,axiom,(
    ! [A_55: pname > $o] :
      ( ! [Y_2: pname] :
          ~ ( member_pname @ Y_2 @ A_55 )
     => ( A_55 = bot_bot_pname_o ) ) )).

thf(fact_532_equals0I,axiom,(
    ! [A_55: hoare_1167836817_state > $o] :
      ( ! [Y_2: hoare_1167836817_state] :
          ~ ( member2058392318_state @ Y_2 @ A_55 )
     => ( A_55 = bot_bo70021908tate_o ) ) )).

thf(fact_533_MGT__alternD,axiom,(
    ! [G_3: hoare_1167836817_state > $o,C_21: com] :
      ( hoare_1201148605gleton
     => ( ( hoare_123228589_state @ G_3
          @ ( insert2134838167_state
            @ ( hoare_908217195_state
              @ ^ [Z_11: state,S0_1: state] :
                ! [S1: state] :
                  ( (=>) @ ( evalc @ C_21 @ S0_1 @ S1 ) @ ( Z_11 = S1 ) )
              @ C_21
              @ fequal_state )
            @ bot_bo70021908tate_o ) )
       => ( hoare_123228589_state @ G_3 @ ( insert2134838167_state @ ( hoare_Mirabelle_MGT @ C_21 ) @ bot_bo70021908tate_o ) ) ) ) )).

thf(fact_534_MGT__alternI,axiom,(
    ! [G_3: hoare_1167836817_state > $o,C_21: com] :
      ( ( hoare_123228589_state @ G_3 @ ( insert2134838167_state @ ( hoare_Mirabelle_MGT @ C_21 ) @ bot_bo70021908tate_o ) )
     => ( hoare_123228589_state @ G_3
        @ ( insert2134838167_state
          @ ( hoare_908217195_state
            @ ^ [Z_11: state,S0_1: state] :
              ! [S1: state] :
                ( (=>) @ ( evalc @ C_21 @ S0_1 @ S1 ) @ ( Z_11 = S1 ) )
            @ C_21
            @ fequal_state )
          @ bot_bo70021908tate_o ) ) ) )).

thf(fact_535_MGT__def,axiom,(
    ! [C_21: com] :
      ( ( hoare_Mirabelle_MGT @ C_21 )
      = ( hoare_908217195_state @ fequal_state @ C_21 @ ( evalc @ C_21 ) ) ) )).

thf(fact_536_evalc_OBody,axiom,(
    ! [Pn_1: pname,S0: state,S1_1: state] :
      ( ( evalc @ ( the_com @ ( body @ Pn_1 ) ) @ S0 @ S1_1 )
     => ( evalc @ ( body_1 @ Pn_1 ) @ S0 @ S1_1 ) ) )).

thf(fact_537_evalc__elim__cases_I6_J,axiom,(
    ! [P: pname,S_2: state,S1_1: state] :
      ( ( evalc @ ( body_1 @ P ) @ S_2 @ S1_1 )
     => ( evalc @ ( the_com @ ( body @ P ) ) @ S_2 @ S1_1 ) ) )).

thf(fact_538_evalc__elim__cases_I1_J,axiom,(
    ! [S_2: state,T: state] :
      ( ( evalc @ skip @ S_2 @ T )
     => ( T = S_2 ) ) )).

thf(fact_539_evalc_OWhileFalse,axiom,(
    ! [C_21: com,B_42: state > $o,S_2: state] :
      ( ~ ( B_42 @ S_2 )
     => ( evalc @ ( while @ B_42 @ C_21 ) @ S_2 @ S_2 ) ) )).

thf(fact_540_evalc_OWhileTrue,axiom,(
    ! [S2: state,C_21: com,S1_1: state,B_42: state > $o,S0: state] :
      ( ( B_42 @ S0 )
     => ( ( evalc @ C_21 @ S0 @ S1_1 )
       => ( ( evalc @ ( while @ B_42 @ C_21 ) @ S1_1 @ S2 )
         => ( evalc @ ( while @ B_42 @ C_21 ) @ S0 @ S2 ) ) ) ) )).

thf(fact_541_evalc_OSemi,axiom,(
    ! [C1: com,S2: state,C0: com,S0: state,S1_1: state] :
      ( ( evalc @ C0 @ S0 @ S1_1 )
     => ( ( evalc @ C1 @ S1_1 @ S2 )
       => ( evalc @ ( semi @ C0 @ C1 ) @ S0 @ S2 ) ) ) )).

thf(fact_542_evalc_OSkip,axiom,(
    ! [S_2: state] :
      ( evalc @ skip @ S_2 @ S_2 ) )).

thf(fact_543_com__det,axiom,(
    ! [U: state,C_21: com,S_2: state,T: state] :
      ( ( evalc @ C_21 @ S_2 @ T )
     => ( ( evalc @ C_21 @ S_2 @ U )
       => ( U = T ) ) ) )).

thf(fact_544_evalc__elim__cases_I4_J,axiom,(
    ! [C1: com,C2: com,S_2: state,T: state] :
      ( ( evalc @ ( semi @ C1 @ C2 ) @ S_2 @ T )
     => ~ ! [S1: state] :
            ( ( evalc @ C1 @ S_2 @ S1 )
           => ~ ( evalc @ C2 @ S1 @ T ) ) ) )).

thf(fact_545_evalc__WHILE__case,axiom,(
    ! [B_42: state > $o,C_21: com,S_2: state,T: state] :
      ( ( evalc @ ( while @ B_42 @ C_21 ) @ S_2 @ T )
     => ( ( ( T = S_2 )
         => ( B_42 @ S_2 ) )
       => ~ ( ( B_42 @ S_2 )
           => ! [S1: state] :
                ( ( evalc @ C_21 @ S_2 @ S1 )
               => ~ ( evalc @ ( while @ B_42 @ C_21 ) @ S1 @ T ) ) ) ) ) )).

thf(fact_546_xt1_I15_J,axiom,(
    ! [C_20: pname > $o,A_54: pname > $o,F_13: ( pname > $o ) > pname > $o,B_41: pname > $o] :
      ( ( A_54
        = ( F_13 @ B_41 ) )
     => ( ( ord_less_eq_pname_o @ C_20 @ B_41 )
       => ( ! [X_5: pname > $o,Y_2: pname > $o] :
              ( ( ord_less_eq_pname_o @ Y_2 @ X_5 )
             => ( ord_less_eq_pname_o @ ( F_13 @ Y_2 ) @ ( F_13 @ X_5 ) ) )
         => ( ord_less_eq_pname_o @ ( F_13 @ C_20 ) @ A_54 ) ) ) ) )).

thf(fact_547_xt1_I15_J,axiom,(
    ! [C_20: hoare_1167836817_state > $o,A_54: hoare_1167836817_state > $o,F_13: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,B_41: hoare_1167836817_state > $o] :
      ( ( A_54
        = ( F_13 @ B_41 ) )
     => ( ( ord_le827224136tate_o @ C_20 @ B_41 )
       => ( ! [X_5: hoare_1167836817_state > $o,Y_2: hoare_1167836817_state > $o] :
              ( ( ord_le827224136tate_o @ Y_2 @ X_5 )
             => ( ord_le827224136tate_o @ ( F_13 @ Y_2 ) @ ( F_13 @ X_5 ) ) )
         => ( ord_le827224136tate_o @ ( F_13 @ C_20 ) @ A_54 ) ) ) ) )).

thf(fact_548_xt1_I16_J,axiom,(
    ! [F_12: ( pname > $o ) > pname > $o,C_19: pname > $o,B_40: pname > $o,A_53: pname > $o] :
      ( ( ord_less_eq_pname_o @ B_40 @ A_53 )
     => ( ( ( F_12 @ B_40 )
          = C_19 )
       => ( ! [X_5: pname > $o,Y_2: pname > $o] :
              ( ( ord_less_eq_pname_o @ Y_2 @ X_5 )
             => ( ord_less_eq_pname_o @ ( F_12 @ Y_2 ) @ ( F_12 @ X_5 ) ) )
         => ( ord_less_eq_pname_o @ C_19 @ ( F_12 @ A_53 ) ) ) ) ) )).

thf(fact_549_xt1_I16_J,axiom,(
    ! [F_12: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,C_19: hoare_1167836817_state > $o,B_40: hoare_1167836817_state > $o,A_53: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ B_40 @ A_53 )
     => ( ( ( F_12 @ B_40 )
          = C_19 )
       => ( ! [X_5: hoare_1167836817_state > $o,Y_2: hoare_1167836817_state > $o] :
              ( ( ord_le827224136tate_o @ Y_2 @ X_5 )
             => ( ord_le827224136tate_o @ ( F_12 @ Y_2 ) @ ( F_12 @ X_5 ) ) )
         => ( ord_le827224136tate_o @ C_19 @ ( F_12 @ A_53 ) ) ) ) ) )).

thf(fact_550_folding__one_Ounion__inter,axiom,(
    ! [B_39: ( pname > $o ) > $o,A_52: ( pname > $o ) > $o,F_11: ( pname > $o ) > ( pname > $o ) > pname > $o,F_10: ( ( pname > $o ) > $o ) > pname > $o] :
      ( ( finite349908348name_o @ F_11 @ F_10 )
     => ( ( finite297249702name_o @ A_52 )
       => ( ( finite297249702name_o @ B_39 )
         => ( ( ( semila2013987940me_o_o @ A_52 @ B_39 )
             != bot_bot_pname_o_o )
           => ( ( F_11 @ ( F_10 @ ( semila181081674me_o_o @ A_52 @ B_39 ) ) @ ( F_10 @ ( semila2013987940me_o_o @ A_52 @ B_39 ) ) )
              = ( F_11 @ ( F_10 @ A_52 ) @ ( F_10 @ B_39 ) ) ) ) ) ) ) )).

thf(fact_551_folding__one_Ounion__inter,axiom,(
    ! [B_39: ( hoare_1167836817_state > $o ) > $o,A_52: ( hoare_1167836817_state > $o ) > $o,F_11: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,F_10: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o] :
      ( ( finite979047547tate_o @ F_11 @ F_10 )
     => ( ( finite1380128977tate_o @ A_52 )
       => ( ( finite1380128977tate_o @ B_39 )
         => ( ( ( semila1758709489te_o_o @ A_52 @ B_39 )
             != bot_bo691907561te_o_o )
           => ( ( F_11 @ ( F_10 @ ( semila866907787te_o_o @ A_52 @ B_39 ) ) @ ( F_10 @ ( semila1758709489te_o_o @ A_52 @ B_39 ) ) )
              = ( F_11 @ ( F_10 @ A_52 ) @ ( F_10 @ B_39 ) ) ) ) ) ) ) )).

thf(fact_552_folding__one_Ounion__inter,axiom,(
    ! [B_39: pname > $o,A_52: pname > $o,F_11: pname > pname > pname,F_10: ( pname > $o ) > pname] :
      ( ( finite1282449217_pname @ F_11 @ F_10 )
     => ( ( finite_finite_pname @ A_52 )
       => ( ( finite_finite_pname @ B_39 )
         => ( ( ( semila1673364395name_o @ A_52 @ B_39 )
             != bot_bot_pname_o )
           => ( ( F_11 @ ( F_10 @ ( semila1780557381name_o @ A_52 @ B_39 ) ) @ ( F_10 @ ( semila1673364395name_o @ A_52 @ B_39 ) ) )
              = ( F_11 @ ( F_10 @ A_52 ) @ ( F_10 @ B_39 ) ) ) ) ) ) ) )).

thf(fact_553_folding__one_Ounion__inter,axiom,(
    ! [B_39: hoare_1167836817_state > $o,A_52: hoare_1167836817_state > $o,F_11: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_10: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
      ( ( finite1074406356_state @ F_11 @ F_10 )
     => ( ( finite1084549118_state @ A_52 )
       => ( ( finite1084549118_state @ B_39 )
         => ( ( ( semila179895820tate_o @ A_52 @ B_39 )
             != bot_bo70021908tate_o )
           => ( ( F_11 @ ( F_10 @ ( semila1172322802tate_o @ A_52 @ B_39 ) ) @ ( F_10 @ ( semila179895820tate_o @ A_52 @ B_39 ) ) )
              = ( F_11 @ ( F_10 @ A_52 ) @ ( F_10 @ B_39 ) ) ) ) ) ) ) )).

thf(fact_554_folding__one_Ounion__disjoint,axiom,(
    ! [B_38: ( pname > $o ) > $o,A_51: ( pname > $o ) > $o,F_9: ( pname > $o ) > ( pname > $o ) > pname > $o,F_8: ( ( pname > $o ) > $o ) > pname > $o] :
      ( ( finite349908348name_o @ F_9 @ F_8 )
     => ( ( finite297249702name_o @ A_51 )
       => ( ( A_51 != bot_bot_pname_o_o )
         => ( ( finite297249702name_o @ B_38 )
           => ( ( B_38 != bot_bot_pname_o_o )
             => ( ( ( semila2013987940me_o_o @ A_51 @ B_38 )
                  = bot_bot_pname_o_o )
               => ( ( F_8 @ ( semila181081674me_o_o @ A_51 @ B_38 ) )
                  = ( F_9 @ ( F_8 @ A_51 ) @ ( F_8 @ B_38 ) ) ) ) ) ) ) ) ) )).

thf(fact_555_folding__one_Ounion__disjoint,axiom,(
    ! [B_38: ( hoare_1167836817_state > $o ) > $o,A_51: ( hoare_1167836817_state > $o ) > $o,F_9: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,F_8: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o] :
      ( ( finite979047547tate_o @ F_9 @ F_8 )
     => ( ( finite1380128977tate_o @ A_51 )
       => ( ( A_51 != bot_bo691907561te_o_o )
         => ( ( finite1380128977tate_o @ B_38 )
           => ( ( B_38 != bot_bo691907561te_o_o )
             => ( ( ( semila1758709489te_o_o @ A_51 @ B_38 )
                  = bot_bo691907561te_o_o )
               => ( ( F_8 @ ( semila866907787te_o_o @ A_51 @ B_38 ) )
                  = ( F_9 @ ( F_8 @ A_51 ) @ ( F_8 @ B_38 ) ) ) ) ) ) ) ) ) )).

thf(fact_556_folding__one_Ounion__disjoint,axiom,(
    ! [B_38: pname > $o,A_51: pname > $o,F_9: pname > pname > pname,F_8: ( pname > $o ) > pname] :
      ( ( finite1282449217_pname @ F_9 @ F_8 )
     => ( ( finite_finite_pname @ A_51 )
       => ( ( A_51 != bot_bot_pname_o )
         => ( ( finite_finite_pname @ B_38 )
           => ( ( B_38 != bot_bot_pname_o )
             => ( ( ( semila1673364395name_o @ A_51 @ B_38 )
                  = bot_bot_pname_o )
               => ( ( F_8 @ ( semila1780557381name_o @ A_51 @ B_38 ) )
                  = ( F_9 @ ( F_8 @ A_51 ) @ ( F_8 @ B_38 ) ) ) ) ) ) ) ) ) )).

thf(fact_557_folding__one_Ounion__disjoint,axiom,(
    ! [B_38: hoare_1167836817_state > $o,A_51: hoare_1167836817_state > $o,F_9: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_8: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
      ( ( finite1074406356_state @ F_9 @ F_8 )
     => ( ( finite1084549118_state @ A_51 )
       => ( ( A_51 != bot_bo70021908tate_o )
         => ( ( finite1084549118_state @ B_38 )
           => ( ( B_38 != bot_bo70021908tate_o )
             => ( ( ( semila179895820tate_o @ A_51 @ B_38 )
                  = bot_bo70021908tate_o )
               => ( ( F_8 @ ( semila1172322802tate_o @ A_51 @ B_38 ) )
                  = ( F_9 @ ( F_8 @ A_51 ) @ ( F_8 @ B_38 ) ) ) ) ) ) ) ) ) )).

thf(fact_558_IntI,axiom,(
    ! [B_37: pname > $o,C_18: pname,A_50: pname > $o] :
      ( ( member_pname @ C_18 @ A_50 )
     => ( ( member_pname @ C_18 @ B_37 )
       => ( member_pname @ C_18 @ ( semila1673364395name_o @ A_50 @ B_37 ) ) ) ) )).

thf(fact_559_IntI,axiom,(
    ! [B_37: hoare_1167836817_state > $o,C_18: hoare_1167836817_state,A_50: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ C_18 @ A_50 )
     => ( ( member2058392318_state @ C_18 @ B_37 )
       => ( member2058392318_state @ C_18 @ ( semila179895820tate_o @ A_50 @ B_37 ) ) ) ) )).

thf(fact_560_IntE,axiom,(
    ! [C_17: pname,A_49: pname > $o,B_36: pname > $o] :
      ( ( member_pname @ C_17 @ ( semila1673364395name_o @ A_49 @ B_36 ) )
     => ~ ( ( member_pname @ C_17 @ A_49 )
         => ~ ( member_pname @ C_17 @ B_36 ) ) ) )).

thf(fact_561_IntE,axiom,(
    ! [C_17: hoare_1167836817_state,A_49: hoare_1167836817_state > $o,B_36: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ C_17 @ ( semila179895820tate_o @ A_49 @ B_36 ) )
     => ~ ( ( member2058392318_state @ C_17 @ A_49 )
         => ~ ( member2058392318_state @ C_17 @ B_36 ) ) ) )).

thf(fact_562_finite__Int,axiom,(
    ! [G_2: ( pname > $o ) > $o,F_7: ( pname > $o ) > $o] :
      ( ( ( finite297249702name_o @ F_7 )
        | ( finite297249702name_o @ G_2 ) )
     => ( finite297249702name_o @ ( semila2013987940me_o_o @ F_7 @ G_2 ) ) ) )).

thf(fact_563_finite__Int,axiom,(
    ! [G_2: ( hoare_1167836817_state > $o ) > $o,F_7: ( hoare_1167836817_state > $o ) > $o] :
      ( ( ( finite1380128977tate_o @ F_7 )
        | ( finite1380128977tate_o @ G_2 ) )
     => ( finite1380128977tate_o @ ( semila1758709489te_o_o @ F_7 @ G_2 ) ) ) )).

thf(fact_564_finite__Int,axiom,(
    ! [G_2: pname > $o,F_7: pname > $o] :
      ( ( ( finite_finite_pname @ F_7 )
        | ( finite_finite_pname @ G_2 ) )
     => ( finite_finite_pname @ ( semila1673364395name_o @ F_7 @ G_2 ) ) ) )).

thf(fact_565_finite__Int,axiom,(
    ! [G_2: hoare_1167836817_state > $o,F_7: hoare_1167836817_state > $o] :
      ( ( ( finite1084549118_state @ F_7 )
        | ( finite1084549118_state @ G_2 ) )
     => ( finite1084549118_state @ ( semila179895820tate_o @ F_7 @ G_2 ) ) ) )).

thf(fact_566_disjoint__iff__not__equal,axiom,(
    ! [A_48: pname > $o,B_35: pname > $o] :
      ( ( ( semila1673364395name_o @ A_48 @ B_35 )
        = bot_bot_pname_o )
    <=> ! [X_5: pname] :
          ( ( member_pname @ X_5 @ A_48 )
         => ! [Xa: pname] :
              ( ( member_pname @ Xa @ B_35 )
             => ( X_5 != Xa ) ) ) ) )).

thf(fact_567_disjoint__iff__not__equal,axiom,(
    ! [A_48: hoare_1167836817_state > $o,B_35: hoare_1167836817_state > $o] :
      ( ( ( semila179895820tate_o @ A_48 @ B_35 )
        = bot_bo70021908tate_o )
    <=> ! [X_5: hoare_1167836817_state] :
          ( ( member2058392318_state @ X_5 @ A_48 )
         => ! [Xa: hoare_1167836817_state] :
              ( ( member2058392318_state @ Xa @ B_35 )
             => ( X_5 != Xa ) ) ) ) )).

thf(fact_568_Int__empty__right,axiom,(
    ! [A_47: pname > $o] :
      ( ( semila1673364395name_o @ A_47 @ bot_bot_pname_o )
      = bot_bot_pname_o ) )).

thf(fact_569_Int__empty__right,axiom,(
    ! [A_47: hoare_1167836817_state > $o] :
      ( ( semila179895820tate_o @ A_47 @ bot_bo70021908tate_o )
      = bot_bo70021908tate_o ) )).

thf(fact_570_Int__empty__left,axiom,(
    ! [B_34: pname > $o] :
      ( ( semila1673364395name_o @ bot_bot_pname_o @ B_34 )
      = bot_bot_pname_o ) )).

thf(fact_571_Int__empty__left,axiom,(
    ! [B_34: hoare_1167836817_state > $o] :
      ( ( semila179895820tate_o @ bot_bo70021908tate_o @ B_34 )
      = bot_bo70021908tate_o ) )).

thf(fact_572_Int__def,axiom,(
    ! [A_46: pname > $o,B_33: pname > $o] :
      ( ( semila1673364395name_o @ A_46 @ B_33 )
      = ( collect_pname
        @ ^ [X_5: pname] :
            ( (&) @ ( member_pname @ X_5 @ A_46 ) @ ( member_pname @ X_5 @ B_33 ) ) ) ) )).

thf(fact_573_Int__def,axiom,(
    ! [A_46: hoare_1167836817_state > $o,B_33: hoare_1167836817_state > $o] :
      ( ( semila179895820tate_o @ A_46 @ B_33 )
      = ( collec1027672124_state
        @ ^ [X_5: hoare_1167836817_state] :
            ( (&) @ ( member2058392318_state @ X_5 @ A_46 ) @ ( member2058392318_state @ X_5 @ B_33 ) ) ) ) )).

thf(fact_574_Int__def,axiom,(
    ! [A_46: ( pname > $o ) > $o,B_33: ( pname > $o ) > $o] :
      ( ( semila2013987940me_o_o @ A_46 @ B_33 )
      = ( collect_pname_o
        @ ^ [X_5: pname > $o] :
            ( (&) @ ( member_pname_o @ X_5 @ A_46 ) @ ( member_pname_o @ X_5 @ B_33 ) ) ) ) )).

thf(fact_575_Int__def,axiom,(
    ! [A_46: ( hoare_1167836817_state > $o ) > $o,B_33: ( hoare_1167836817_state > $o ) > $o] :
      ( ( semila1758709489te_o_o @ A_46 @ B_33 )
      = ( collec269976083tate_o
        @ ^ [X_5: hoare_1167836817_state > $o] :
            ( (&) @ ( member864234961tate_o @ X_5 @ A_46 ) @ ( member864234961tate_o @ X_5 @ B_33 ) ) ) ) )).

thf(fact_576_Int__iff,axiom,(
    ! [C_16: pname,A_45: pname > $o,B_32: pname > $o] :
      ( ( member_pname @ C_16 @ ( semila1673364395name_o @ A_45 @ B_32 ) )
    <=> ( ( member_pname @ C_16 @ A_45 )
        & ( member_pname @ C_16 @ B_32 ) ) ) )).

thf(fact_577_Int__iff,axiom,(
    ! [C_16: hoare_1167836817_state,A_45: hoare_1167836817_state > $o,B_32: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ C_16 @ ( semila179895820tate_o @ A_45 @ B_32 ) )
    <=> ( ( member2058392318_state @ C_16 @ A_45 )
        & ( member2058392318_state @ C_16 @ B_32 ) ) ) )).

thf(fact_578_IntD1,axiom,(
    ! [C_15: pname,A_44: pname > $o,B_31: pname > $o] :
      ( ( member_pname @ C_15 @ ( semila1673364395name_o @ A_44 @ B_31 ) )
     => ( member_pname @ C_15 @ A_44 ) ) )).

thf(fact_579_IntD1,axiom,(
    ! [C_15: hoare_1167836817_state,A_44: hoare_1167836817_state > $o,B_31: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ C_15 @ ( semila179895820tate_o @ A_44 @ B_31 ) )
     => ( member2058392318_state @ C_15 @ A_44 ) ) )).

thf(fact_580_IntD2,axiom,(
    ! [C_14: pname,A_43: pname > $o,B_30: pname > $o] :
      ( ( member_pname @ C_14 @ ( semila1673364395name_o @ A_43 @ B_30 ) )
     => ( member_pname @ C_14 @ B_30 ) ) )).

thf(fact_581_IntD2,axiom,(
    ! [C_14: hoare_1167836817_state,A_43: hoare_1167836817_state > $o,B_30: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ C_14 @ ( semila179895820tate_o @ A_43 @ B_30 ) )
     => ( member2058392318_state @ C_14 @ B_30 ) ) )).

thf(fact_582_Collect__conj__eq,axiom,(
    ! [P_5: hoare_1167836817_state > $o,Q_1: hoare_1167836817_state > $o] :
      ( ( collec1027672124_state
        @ ^ [X_5: hoare_1167836817_state] :
            ( (&) @ ( P_5 @ X_5 ) @ ( Q_1 @ X_5 ) ) )
      = ( semila179895820tate_o @ ( collec1027672124_state @ P_5 ) @ ( collec1027672124_state @ Q_1 ) ) ) )).

thf(fact_583_Collect__conj__eq,axiom,(
    ! [P_5: pname > $o,Q_1: pname > $o] :
      ( ( collect_pname
        @ ^ [X_5: pname] :
            ( (&) @ ( P_5 @ X_5 ) @ ( Q_1 @ X_5 ) ) )
      = ( semila1673364395name_o @ ( collect_pname @ P_5 ) @ ( collect_pname @ Q_1 ) ) ) )).

thf(fact_584_Collect__conj__eq,axiom,(
    ! [P_5: ( pname > $o ) > $o,Q_1: ( pname > $o ) > $o] :
      ( ( collect_pname_o
        @ ^ [X_5: pname > $o] :
            ( (&) @ ( P_5 @ X_5 ) @ ( Q_1 @ X_5 ) ) )
      = ( semila2013987940me_o_o @ ( collect_pname_o @ P_5 ) @ ( collect_pname_o @ Q_1 ) ) ) )).

thf(fact_585_Collect__conj__eq,axiom,(
    ! [P_5: ( hoare_1167836817_state > $o ) > $o,Q_1: ( hoare_1167836817_state > $o ) > $o] :
      ( ( collec269976083tate_o
        @ ^ [X_5: hoare_1167836817_state > $o] :
            ( (&) @ ( P_5 @ X_5 ) @ ( Q_1 @ X_5 ) ) )
      = ( semila1758709489te_o_o @ ( collec269976083tate_o @ P_5 ) @ ( collec269976083tate_o @ Q_1 ) ) ) )).

thf(fact_586_Int__Collect,axiom,(
    ! [X_30: pname,A_42: pname > $o,P_4: pname > $o] :
      ( ( member_pname @ X_30 @ ( semila1673364395name_o @ A_42 @ ( collect_pname @ P_4 ) ) )
    <=> ( ( member_pname @ X_30 @ A_42 )
        & ( P_4 @ X_30 ) ) ) )).

thf(fact_587_Int__Collect,axiom,(
    ! [X_30: hoare_1167836817_state,A_42: hoare_1167836817_state > $o,P_4: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ X_30 @ ( semila179895820tate_o @ A_42 @ ( collec1027672124_state @ P_4 ) ) )
    <=> ( ( member2058392318_state @ X_30 @ A_42 )
        & ( P_4 @ X_30 ) ) ) )).

thf(fact_588_Int__Collect,axiom,(
    ! [X_30: pname > $o,A_42: ( pname > $o ) > $o,P_4: ( pname > $o ) > $o] :
      ( ( member_pname_o @ X_30 @ ( semila2013987940me_o_o @ A_42 @ ( collect_pname_o @ P_4 ) ) )
    <=> ( ( member_pname_o @ X_30 @ A_42 )
        & ( P_4 @ X_30 ) ) ) )).

thf(fact_589_Int__Collect,axiom,(
    ! [X_30: hoare_1167836817_state > $o,A_42: ( hoare_1167836817_state > $o ) > $o,P_4: ( hoare_1167836817_state > $o ) > $o] :
      ( ( member864234961tate_o @ X_30 @ ( semila1758709489te_o_o @ A_42 @ ( collec269976083tate_o @ P_4 ) ) )
    <=> ( ( member864234961tate_o @ X_30 @ A_42 )
        & ( P_4 @ X_30 ) ) ) )).

thf(fact_590_inf__Int__eq,axiom,(
    ! [R: pname > $o,S_1: pname > $o,X_5: pname] :
      ( ( semila1673364395name_o
        @ ^ [Y_2: pname] :
            ( member_pname @ Y_2 @ R )
        @ ^ [Y_2: pname] :
            ( member_pname @ Y_2 @ S_1 )
        @ X_5 )
    <=> ( member_pname @ X_5 @ ( semila1673364395name_o @ R @ S_1 ) ) ) )).

thf(fact_591_inf__Int__eq,axiom,(
    ! [R: hoare_1167836817_state > $o,S_1: hoare_1167836817_state > $o,X_5: hoare_1167836817_state] :
      ( ( semila179895820tate_o
        @ ^ [Y_2: hoare_1167836817_state] :
            ( member2058392318_state @ Y_2 @ R )
        @ ^ [Y_2: hoare_1167836817_state] :
            ( member2058392318_state @ Y_2 @ S_1 )
        @ X_5 )
    <=> ( member2058392318_state @ X_5 @ ( semila179895820tate_o @ R @ S_1 ) ) ) )).

thf(fact_592_Un__Int__crazy,axiom,(
    ! [A_41: hoare_1167836817_state > $o,B_29: hoare_1167836817_state > $o,C_13: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ ( semila1172322802tate_o @ ( semila179895820tate_o @ A_41 @ B_29 ) @ ( semila179895820tate_o @ B_29 @ C_13 ) ) @ ( semila179895820tate_o @ C_13 @ A_41 ) )
      = ( semila179895820tate_o @ ( semila179895820tate_o @ ( semila1172322802tate_o @ A_41 @ B_29 ) @ ( semila1172322802tate_o @ B_29 @ C_13 ) ) @ ( semila1172322802tate_o @ C_13 @ A_41 ) ) ) )).

thf(fact_593_Un__Int__distrib2,axiom,(
    ! [B_28: hoare_1167836817_state > $o,C_12: hoare_1167836817_state > $o,A_40: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ ( semila179895820tate_o @ B_28 @ C_12 ) @ A_40 )
      = ( semila179895820tate_o @ ( semila1172322802tate_o @ B_28 @ A_40 ) @ ( semila1172322802tate_o @ C_12 @ A_40 ) ) ) )).

thf(fact_594_Int__Un__distrib2,axiom,(
    ! [B_27: hoare_1167836817_state > $o,C_11: hoare_1167836817_state > $o,A_39: hoare_1167836817_state > $o] :
      ( ( semila179895820tate_o @ ( semila1172322802tate_o @ B_27 @ C_11 ) @ A_39 )
      = ( semila1172322802tate_o @ ( semila179895820tate_o @ B_27 @ A_39 ) @ ( semila179895820tate_o @ C_11 @ A_39 ) ) ) )).

thf(fact_595_Un__Int__distrib,axiom,(
    ! [A_38: hoare_1167836817_state > $o,B_26: hoare_1167836817_state > $o,C_10: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ A_38 @ ( semila179895820tate_o @ B_26 @ C_10 ) )
      = ( semila179895820tate_o @ ( semila1172322802tate_o @ A_38 @ B_26 ) @ ( semila1172322802tate_o @ A_38 @ C_10 ) ) ) )).

thf(fact_596_Int__Un__distrib,axiom,(
    ! [A_37: hoare_1167836817_state > $o,B_25: hoare_1167836817_state > $o,C_9: hoare_1167836817_state > $o] :
      ( ( semila179895820tate_o @ A_37 @ ( semila1172322802tate_o @ B_25 @ C_9 ) )
      = ( semila1172322802tate_o @ ( semila179895820tate_o @ A_37 @ B_25 ) @ ( semila179895820tate_o @ A_37 @ C_9 ) ) ) )).

thf(fact_597_Int__mono,axiom,(
    ! [B_24: pname > $o,D_1: pname > $o,A_36: pname > $o,C_8: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_36 @ C_8 )
     => ( ( ord_less_eq_pname_o @ B_24 @ D_1 )
       => ( ord_less_eq_pname_o @ ( semila1673364395name_o @ A_36 @ B_24 ) @ ( semila1673364395name_o @ C_8 @ D_1 ) ) ) ) )).

thf(fact_598_Int__mono,axiom,(
    ! [B_24: hoare_1167836817_state > $o,D_1: hoare_1167836817_state > $o,A_36: hoare_1167836817_state > $o,C_8: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_36 @ C_8 )
     => ( ( ord_le827224136tate_o @ B_24 @ D_1 )
       => ( ord_le827224136tate_o @ ( semila179895820tate_o @ A_36 @ B_24 ) @ ( semila179895820tate_o @ C_8 @ D_1 ) ) ) ) )).

thf(fact_599_Int__greatest,axiom,(
    ! [B_23: pname > $o,C_7: pname > $o,A_35: pname > $o] :
      ( ( ord_less_eq_pname_o @ C_7 @ A_35 )
     => ( ( ord_less_eq_pname_o @ C_7 @ B_23 )
       => ( ord_less_eq_pname_o @ C_7 @ ( semila1673364395name_o @ A_35 @ B_23 ) ) ) ) )).

thf(fact_600_Int__greatest,axiom,(
    ! [B_23: hoare_1167836817_state > $o,C_7: hoare_1167836817_state > $o,A_35: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ C_7 @ A_35 )
     => ( ( ord_le827224136tate_o @ C_7 @ B_23 )
       => ( ord_le827224136tate_o @ C_7 @ ( semila179895820tate_o @ A_35 @ B_23 ) ) ) ) )).

thf(fact_601_Int__absorb1,axiom,(
    ! [B_22: pname > $o,A_34: pname > $o] :
      ( ( ord_less_eq_pname_o @ B_22 @ A_34 )
     => ( ( semila1673364395name_o @ A_34 @ B_22 )
        = B_22 ) ) )).

thf(fact_602_Int__absorb1,axiom,(
    ! [B_22: hoare_1167836817_state > $o,A_34: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ B_22 @ A_34 )
     => ( ( semila179895820tate_o @ A_34 @ B_22 )
        = B_22 ) ) )).

thf(fact_603_Int__absorb2,axiom,(
    ! [A_33: pname > $o,B_21: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_33 @ B_21 )
     => ( ( semila1673364395name_o @ A_33 @ B_21 )
        = A_33 ) ) )).

thf(fact_604_Int__absorb2,axiom,(
    ! [A_33: hoare_1167836817_state > $o,B_21: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_33 @ B_21 )
     => ( ( semila179895820tate_o @ A_33 @ B_21 )
        = A_33 ) ) )).

thf(fact_605_Int__lower2,axiom,(
    ! [A_32: pname > $o,B_20: pname > $o] :
      ( ord_less_eq_pname_o @ ( semila1673364395name_o @ A_32 @ B_20 ) @ B_20 ) )).

thf(fact_606_Int__lower2,axiom,(
    ! [A_32: hoare_1167836817_state > $o,B_20: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ ( semila179895820tate_o @ A_32 @ B_20 ) @ B_20 ) )).

thf(fact_607_Int__lower1,axiom,(
    ! [A_31: pname > $o,B_19: pname > $o] :
      ( ord_less_eq_pname_o @ ( semila1673364395name_o @ A_31 @ B_19 ) @ A_31 ) )).

thf(fact_608_Int__lower1,axiom,(
    ! [A_31: hoare_1167836817_state > $o,B_19: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ ( semila179895820tate_o @ A_31 @ B_19 ) @ A_31 ) )).

thf(fact_609_Int__insert__left__if1,axiom,(
    ! [B_18: pname > $o,A_30: pname,C_6: pname > $o] :
      ( ( member_pname @ A_30 @ C_6 )
     => ( ( semila1673364395name_o @ ( insert_pname @ A_30 @ B_18 ) @ C_6 )
        = ( insert_pname @ A_30 @ ( semila1673364395name_o @ B_18 @ C_6 ) ) ) ) )).

thf(fact_610_Int__insert__left__if1,axiom,(
    ! [B_18: hoare_1167836817_state > $o,A_30: hoare_1167836817_state,C_6: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ A_30 @ C_6 )
     => ( ( semila179895820tate_o @ ( insert2134838167_state @ A_30 @ B_18 ) @ C_6 )
        = ( insert2134838167_state @ A_30 @ ( semila179895820tate_o @ B_18 @ C_6 ) ) ) ) )).

thf(fact_611_Int__insert__right__if1,axiom,(
    ! [B_17: pname > $o,A_29: pname,A_28: pname > $o] :
      ( ( member_pname @ A_29 @ A_28 )
     => ( ( semila1673364395name_o @ A_28 @ ( insert_pname @ A_29 @ B_17 ) )
        = ( insert_pname @ A_29 @ ( semila1673364395name_o @ A_28 @ B_17 ) ) ) ) )).

thf(fact_612_Int__insert__right__if1,axiom,(
    ! [B_17: hoare_1167836817_state > $o,A_29: hoare_1167836817_state,A_28: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ A_29 @ A_28 )
     => ( ( semila179895820tate_o @ A_28 @ ( insert2134838167_state @ A_29 @ B_17 ) )
        = ( insert2134838167_state @ A_29 @ ( semila179895820tate_o @ A_28 @ B_17 ) ) ) ) )).

thf(fact_613_Int__insert__left__if0,axiom,(
    ! [B_16: pname > $o,A_27: pname,C_5: pname > $o] :
      ( ~ ( member_pname @ A_27 @ C_5 )
     => ( ( semila1673364395name_o @ ( insert_pname @ A_27 @ B_16 ) @ C_5 )
        = ( semila1673364395name_o @ B_16 @ C_5 ) ) ) )).

thf(fact_614_Int__insert__left__if0,axiom,(
    ! [B_16: hoare_1167836817_state > $o,A_27: hoare_1167836817_state,C_5: hoare_1167836817_state > $o] :
      ( ~ ( member2058392318_state @ A_27 @ C_5 )
     => ( ( semila179895820tate_o @ ( insert2134838167_state @ A_27 @ B_16 ) @ C_5 )
        = ( semila179895820tate_o @ B_16 @ C_5 ) ) ) )).

thf(fact_615_Int__insert__right__if0,axiom,(
    ! [B_15: pname > $o,A_26: pname,A_25: pname > $o] :
      ( ~ ( member_pname @ A_26 @ A_25 )
     => ( ( semila1673364395name_o @ A_25 @ ( insert_pname @ A_26 @ B_15 ) )
        = ( semila1673364395name_o @ A_25 @ B_15 ) ) ) )).

thf(fact_616_Int__insert__right__if0,axiom,(
    ! [B_15: hoare_1167836817_state > $o,A_26: hoare_1167836817_state,A_25: hoare_1167836817_state > $o] :
      ( ~ ( member2058392318_state @ A_26 @ A_25 )
     => ( ( semila179895820tate_o @ A_25 @ ( insert2134838167_state @ A_26 @ B_15 ) )
        = ( semila179895820tate_o @ A_25 @ B_15 ) ) ) )).

thf(fact_617_insert__inter__insert,axiom,(
    ! [A_24: pname,A_23: pname > $o,B_14: pname > $o] :
      ( ( semila1673364395name_o @ ( insert_pname @ A_24 @ A_23 ) @ ( insert_pname @ A_24 @ B_14 ) )
      = ( insert_pname @ A_24 @ ( semila1673364395name_o @ A_23 @ B_14 ) ) ) )).

thf(fact_618_insert__inter__insert,axiom,(
    ! [A_24: hoare_1167836817_state,A_23: hoare_1167836817_state > $o,B_14: hoare_1167836817_state > $o] :
      ( ( semila179895820tate_o @ ( insert2134838167_state @ A_24 @ A_23 ) @ ( insert2134838167_state @ A_24 @ B_14 ) )
      = ( insert2134838167_state @ A_24 @ ( semila179895820tate_o @ A_23 @ B_14 ) ) ) )).

thf(fact_619_Int__insert__left,axiom,(
    ! [B_13: pname > $o,A_22: pname,C_4: pname > $o] :
      ( ( ( member_pname @ A_22 @ C_4 )
       => ( ( semila1673364395name_o @ ( insert_pname @ A_22 @ B_13 ) @ C_4 )
          = ( insert_pname @ A_22 @ ( semila1673364395name_o @ B_13 @ C_4 ) ) ) )
      & ( ~ ( member_pname @ A_22 @ C_4 )
       => ( ( semila1673364395name_o @ ( insert_pname @ A_22 @ B_13 ) @ C_4 )
          = ( semila1673364395name_o @ B_13 @ C_4 ) ) ) ) )).

thf(fact_620_Int__insert__left,axiom,(
    ! [B_13: hoare_1167836817_state > $o,A_22: hoare_1167836817_state,C_4: hoare_1167836817_state > $o] :
      ( ( ( member2058392318_state @ A_22 @ C_4 )
       => ( ( semila179895820tate_o @ ( insert2134838167_state @ A_22 @ B_13 ) @ C_4 )
          = ( insert2134838167_state @ A_22 @ ( semila179895820tate_o @ B_13 @ C_4 ) ) ) )
      & ( ~ ( member2058392318_state @ A_22 @ C_4 )
       => ( ( semila179895820tate_o @ ( insert2134838167_state @ A_22 @ B_13 ) @ C_4 )
          = ( semila179895820tate_o @ B_13 @ C_4 ) ) ) ) )).

thf(fact_621_Int__insert__right,axiom,(
    ! [B_12: pname > $o,A_21: pname,A_20: pname > $o] :
      ( ( ( member_pname @ A_21 @ A_20 )
       => ( ( semila1673364395name_o @ A_20 @ ( insert_pname @ A_21 @ B_12 ) )
          = ( insert_pname @ A_21 @ ( semila1673364395name_o @ A_20 @ B_12 ) ) ) )
      & ( ~ ( member_pname @ A_21 @ A_20 )
       => ( ( semila1673364395name_o @ A_20 @ ( insert_pname @ A_21 @ B_12 ) )
          = ( semila1673364395name_o @ A_20 @ B_12 ) ) ) ) )).

thf(fact_622_Int__insert__right,axiom,(
    ! [B_12: hoare_1167836817_state > $o,A_21: hoare_1167836817_state,A_20: hoare_1167836817_state > $o] :
      ( ( ( member2058392318_state @ A_21 @ A_20 )
       => ( ( semila179895820tate_o @ A_20 @ ( insert2134838167_state @ A_21 @ B_12 ) )
          = ( insert2134838167_state @ A_21 @ ( semila179895820tate_o @ A_20 @ B_12 ) ) ) )
      & ( ~ ( member2058392318_state @ A_21 @ A_20 )
       => ( ( semila179895820tate_o @ A_20 @ ( insert2134838167_state @ A_21 @ B_12 ) )
          = ( semila179895820tate_o @ A_20 @ B_12 ) ) ) ) )).

thf(fact_623_inf__sup__ord_I1_J,axiom,(
    ! [X_29: pname > $o,Y_20: pname > $o] :
      ( ord_less_eq_pname_o @ ( semila1673364395name_o @ X_29 @ Y_20 ) @ X_29 ) )).

thf(fact_624_inf__sup__ord_I1_J,axiom,(
    ! [X_29: hoare_1167836817_state > $o,Y_20: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ ( semila179895820tate_o @ X_29 @ Y_20 ) @ X_29 ) )).

thf(fact_625_inf__le1,axiom,(
    ! [X_28: pname > $o,Y_19: pname > $o] :
      ( ord_less_eq_pname_o @ ( semila1673364395name_o @ X_28 @ Y_19 ) @ X_28 ) )).

thf(fact_626_inf__le1,axiom,(
    ! [X_28: hoare_1167836817_state > $o,Y_19: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ ( semila179895820tate_o @ X_28 @ Y_19 ) @ X_28 ) )).

thf(fact_627_inf__sup__ord_I2_J,axiom,(
    ! [X_27: pname > $o,Y_18: pname > $o] :
      ( ord_less_eq_pname_o @ ( semila1673364395name_o @ X_27 @ Y_18 ) @ Y_18 ) )).

thf(fact_628_inf__sup__ord_I2_J,axiom,(
    ! [X_27: hoare_1167836817_state > $o,Y_18: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ ( semila179895820tate_o @ X_27 @ Y_18 ) @ Y_18 ) )).

thf(fact_629_inf__le2,axiom,(
    ! [X_26: pname > $o,Y_17: pname > $o] :
      ( ord_less_eq_pname_o @ ( semila1673364395name_o @ X_26 @ Y_17 ) @ Y_17 ) )).

thf(fact_630_inf__le2,axiom,(
    ! [X_26: hoare_1167836817_state > $o,Y_17: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ ( semila179895820tate_o @ X_26 @ Y_17 ) @ Y_17 ) )).

thf(fact_631_le__iff__inf,axiom,(
    ! [X_25: pname > $o,Y_16: pname > $o] :
      ( ( ord_less_eq_pname_o @ X_25 @ Y_16 )
    <=> ( ( semila1673364395name_o @ X_25 @ Y_16 )
        = X_25 ) ) )).

thf(fact_632_le__iff__inf,axiom,(
    ! [X_25: hoare_1167836817_state > $o,Y_16: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ X_25 @ Y_16 )
    <=> ( ( semila179895820tate_o @ X_25 @ Y_16 )
        = X_25 ) ) )).

thf(fact_633_le__inf__iff,axiom,(
    ! [X_24: pname > $o,Y_15: pname > $o,Z_10: pname > $o] :
      ( ( ord_less_eq_pname_o @ X_24 @ ( semila1673364395name_o @ Y_15 @ Z_10 ) )
    <=> ( ( ord_less_eq_pname_o @ X_24 @ Y_15 )
        & ( ord_less_eq_pname_o @ X_24 @ Z_10 ) ) ) )).

thf(fact_634_le__inf__iff,axiom,(
    ! [X_24: hoare_1167836817_state > $o,Y_15: hoare_1167836817_state > $o,Z_10: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ X_24 @ ( semila179895820tate_o @ Y_15 @ Z_10 ) )
    <=> ( ( ord_le827224136tate_o @ X_24 @ Y_15 )
        & ( ord_le827224136tate_o @ X_24 @ Z_10 ) ) ) )).

thf(fact_635_le__infI1,axiom,(
    ! [B_11: pname > $o,A_19: pname > $o,X_23: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_19 @ X_23 )
     => ( ord_less_eq_pname_o @ ( semila1673364395name_o @ A_19 @ B_11 ) @ X_23 ) ) )).

thf(fact_636_le__infI1,axiom,(
    ! [B_11: hoare_1167836817_state > $o,A_19: hoare_1167836817_state > $o,X_23: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_19 @ X_23 )
     => ( ord_le827224136tate_o @ ( semila179895820tate_o @ A_19 @ B_11 ) @ X_23 ) ) )).

thf(fact_637_le__infI2,axiom,(
    ! [A_18: pname > $o,B_10: pname > $o,X_22: pname > $o] :
      ( ( ord_less_eq_pname_o @ B_10 @ X_22 )
     => ( ord_less_eq_pname_o @ ( semila1673364395name_o @ A_18 @ B_10 ) @ X_22 ) ) )).

thf(fact_638_le__infI2,axiom,(
    ! [A_18: hoare_1167836817_state > $o,B_10: hoare_1167836817_state > $o,X_22: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ B_10 @ X_22 )
     => ( ord_le827224136tate_o @ ( semila179895820tate_o @ A_18 @ B_10 ) @ X_22 ) ) )).

thf(fact_639_inf__absorb1,axiom,(
    ! [X_21: pname > $o,Y_14: pname > $o] :
      ( ( ord_less_eq_pname_o @ X_21 @ Y_14 )
     => ( ( semila1673364395name_o @ X_21 @ Y_14 )
        = X_21 ) ) )).

thf(fact_640_inf__absorb1,axiom,(
    ! [X_21: hoare_1167836817_state > $o,Y_14: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ X_21 @ Y_14 )
     => ( ( semila179895820tate_o @ X_21 @ Y_14 )
        = X_21 ) ) )).

thf(fact_641_inf__absorb2,axiom,(
    ! [Y_13: pname > $o,X_20: pname > $o] :
      ( ( ord_less_eq_pname_o @ Y_13 @ X_20 )
     => ( ( semila1673364395name_o @ X_20 @ Y_13 )
        = Y_13 ) ) )).

thf(fact_642_inf__absorb2,axiom,(
    ! [Y_13: hoare_1167836817_state > $o,X_20: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ Y_13 @ X_20 )
     => ( ( semila179895820tate_o @ X_20 @ Y_13 )
        = Y_13 ) ) )).

thf(fact_643_le__infI,axiom,(
    ! [B_9: pname > $o,X_19: pname > $o,A_17: pname > $o] :
      ( ( ord_less_eq_pname_o @ X_19 @ A_17 )
     => ( ( ord_less_eq_pname_o @ X_19 @ B_9 )
       => ( ord_less_eq_pname_o @ X_19 @ ( semila1673364395name_o @ A_17 @ B_9 ) ) ) ) )).

thf(fact_644_le__infI,axiom,(
    ! [B_9: hoare_1167836817_state > $o,X_19: hoare_1167836817_state > $o,A_17: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ X_19 @ A_17 )
     => ( ( ord_le827224136tate_o @ X_19 @ B_9 )
       => ( ord_le827224136tate_o @ X_19 @ ( semila179895820tate_o @ A_17 @ B_9 ) ) ) ) )).

thf(fact_645_inf__greatest,axiom,(
    ! [Z_9: pname > $o,X_18: pname > $o,Y_12: pname > $o] :
      ( ( ord_less_eq_pname_o @ X_18 @ Y_12 )
     => ( ( ord_less_eq_pname_o @ X_18 @ Z_9 )
       => ( ord_less_eq_pname_o @ X_18 @ ( semila1673364395name_o @ Y_12 @ Z_9 ) ) ) ) )).

thf(fact_646_inf__greatest,axiom,(
    ! [Z_9: hoare_1167836817_state > $o,X_18: hoare_1167836817_state > $o,Y_12: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ X_18 @ Y_12 )
     => ( ( ord_le827224136tate_o @ X_18 @ Z_9 )
       => ( ord_le827224136tate_o @ X_18 @ ( semila179895820tate_o @ Y_12 @ Z_9 ) ) ) ) )).

thf(fact_647_inf__mono,axiom,(
    ! [B_8: pname > $o,D: pname > $o,A_16: pname > $o,C_3: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_16 @ C_3 )
     => ( ( ord_less_eq_pname_o @ B_8 @ D )
       => ( ord_less_eq_pname_o @ ( semila1673364395name_o @ A_16 @ B_8 ) @ ( semila1673364395name_o @ C_3 @ D ) ) ) ) )).

thf(fact_648_inf__mono,axiom,(
    ! [B_8: hoare_1167836817_state > $o,D: hoare_1167836817_state > $o,A_16: hoare_1167836817_state > $o,C_3: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_16 @ C_3 )
     => ( ( ord_le827224136tate_o @ B_8 @ D )
       => ( ord_le827224136tate_o @ ( semila179895820tate_o @ A_16 @ B_8 ) @ ( semila179895820tate_o @ C_3 @ D ) ) ) ) )).

thf(fact_649_le__infE,axiom,(
    ! [X_17: pname > $o,A_15: pname > $o,B_7: pname > $o] :
      ( ( ord_less_eq_pname_o @ X_17 @ ( semila1673364395name_o @ A_15 @ B_7 ) )
     => ~ ( ( ord_less_eq_pname_o @ X_17 @ A_15 )
         => ~ ( ord_less_eq_pname_o @ X_17 @ B_7 ) ) ) )).

thf(fact_650_le__infE,axiom,(
    ! [X_17: hoare_1167836817_state > $o,A_15: hoare_1167836817_state > $o,B_7: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ X_17 @ ( semila179895820tate_o @ A_15 @ B_7 ) )
     => ~ ( ( ord_le827224136tate_o @ X_17 @ A_15 )
         => ~ ( ord_le827224136tate_o @ X_17 @ B_7 ) ) ) )).

thf(fact_651_inf__sup__absorb,axiom,(
    ! [X_16: hoare_1167836817_state > $o,Y_11: hoare_1167836817_state > $o] :
      ( ( semila179895820tate_o @ X_16 @ ( semila1172322802tate_o @ X_16 @ Y_11 ) )
      = X_16 ) )).

thf(fact_652_sup__inf__absorb,axiom,(
    ! [X_15: hoare_1167836817_state > $o,Y_10: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ X_15 @ ( semila179895820tate_o @ X_15 @ Y_10 ) )
      = X_15 ) )).

thf(fact_653_inf__sup__distrib1,axiom,(
    ! [X_14: hoare_1167836817_state > $o,Y_9: hoare_1167836817_state > $o,Z_8: hoare_1167836817_state > $o] :
      ( ( semila179895820tate_o @ X_14 @ ( semila1172322802tate_o @ Y_9 @ Z_8 ) )
      = ( semila1172322802tate_o @ ( semila179895820tate_o @ X_14 @ Y_9 ) @ ( semila179895820tate_o @ X_14 @ Z_8 ) ) ) )).

thf(fact_654_sup__inf__distrib1,axiom,(
    ! [X_13: hoare_1167836817_state > $o,Y_8: hoare_1167836817_state > $o,Z_7: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ X_13 @ ( semila179895820tate_o @ Y_8 @ Z_7 ) )
      = ( semila179895820tate_o @ ( semila1172322802tate_o @ X_13 @ Y_8 ) @ ( semila1172322802tate_o @ X_13 @ Z_7 ) ) ) )).

thf(fact_655_inf__sup__distrib2,axiom,(
    ! [Y_7: hoare_1167836817_state > $o,Z_6: hoare_1167836817_state > $o,X_12: hoare_1167836817_state > $o] :
      ( ( semila179895820tate_o @ ( semila1172322802tate_o @ Y_7 @ Z_6 ) @ X_12 )
      = ( semila1172322802tate_o @ ( semila179895820tate_o @ Y_7 @ X_12 ) @ ( semila179895820tate_o @ Z_6 @ X_12 ) ) ) )).

thf(fact_656_sup__inf__distrib2,axiom,(
    ! [Y_6: hoare_1167836817_state > $o,Z_5: hoare_1167836817_state > $o,X_11: hoare_1167836817_state > $o] :
      ( ( semila1172322802tate_o @ ( semila179895820tate_o @ Y_6 @ Z_5 ) @ X_11 )
      = ( semila179895820tate_o @ ( semila1172322802tate_o @ Y_6 @ X_11 ) @ ( semila1172322802tate_o @ Z_5 @ X_11 ) ) ) )).

thf(fact_657_inf__bot__right,axiom,(
    ! [X_10: pname > $o] :
      ( ( semila1673364395name_o @ X_10 @ bot_bot_pname_o )
      = bot_bot_pname_o ) )).

thf(fact_658_inf__bot__right,axiom,(
    ! [X_10: hoare_1167836817_state > $o] :
      ( ( semila179895820tate_o @ X_10 @ bot_bo70021908tate_o )
      = bot_bo70021908tate_o ) )).

thf(fact_659_inf__bot__left,axiom,(
    ! [X_9: pname > $o] :
      ( ( semila1673364395name_o @ bot_bot_pname_o @ X_9 )
      = bot_bot_pname_o ) )).

thf(fact_660_inf__bot__left,axiom,(
    ! [X_9: hoare_1167836817_state > $o] :
      ( ( semila179895820tate_o @ bot_bo70021908tate_o @ X_9 )
      = bot_bo70021908tate_o ) )).

thf(fact_661_distrib__sup__le,axiom,(
    ! [X_8: pname > $o,Y_5: pname > $o,Z_4: pname > $o] :
      ( ord_less_eq_pname_o @ ( semila1780557381name_o @ X_8 @ ( semila1673364395name_o @ Y_5 @ Z_4 ) ) @ ( semila1673364395name_o @ ( semila1780557381name_o @ X_8 @ Y_5 ) @ ( semila1780557381name_o @ X_8 @ Z_4 ) ) ) )).

thf(fact_662_distrib__sup__le,axiom,(
    ! [X_8: hoare_1167836817_state > $o,Y_5: hoare_1167836817_state > $o,Z_4: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ ( semila1172322802tate_o @ X_8 @ ( semila179895820tate_o @ Y_5 @ Z_4 ) ) @ ( semila179895820tate_o @ ( semila1172322802tate_o @ X_8 @ Y_5 ) @ ( semila1172322802tate_o @ X_8 @ Z_4 ) ) ) )).

thf(fact_663_distrib__inf__le,axiom,(
    ! [X_7: pname > $o,Y_4: pname > $o,Z_3: pname > $o] :
      ( ord_less_eq_pname_o @ ( semila1780557381name_o @ ( semila1673364395name_o @ X_7 @ Y_4 ) @ ( semila1673364395name_o @ X_7 @ Z_3 ) ) @ ( semila1673364395name_o @ X_7 @ ( semila1780557381name_o @ Y_4 @ Z_3 ) ) ) )).

thf(fact_664_distrib__inf__le,axiom,(
    ! [X_7: hoare_1167836817_state > $o,Y_4: hoare_1167836817_state > $o,Z_3: hoare_1167836817_state > $o] :
      ( ord_le827224136tate_o @ ( semila1172322802tate_o @ ( semila179895820tate_o @ X_7 @ Y_4 ) @ ( semila179895820tate_o @ X_7 @ Z_3 ) ) @ ( semila179895820tate_o @ X_7 @ ( semila1172322802tate_o @ Y_4 @ Z_3 ) ) ) )).

thf(fact_665_image__Int__subset,axiom,(
    ! [F_6: pname > hoare_1167836817_state,A_14: pname > $o,B_6: pname > $o] :
      ( ord_le827224136tate_o @ ( image_575578384_state @ F_6 @ ( semila1673364395name_o @ A_14 @ B_6 ) ) @ ( semila179895820tate_o @ ( image_575578384_state @ F_6 @ A_14 ) @ ( image_575578384_state @ F_6 @ B_6 ) ) ) )).

thf(fact_666_Un__Int__assoc__eq,axiom,(
    ! [A_13: pname > $o,B_5: pname > $o,C_2: pname > $o] :
      ( ( ( semila1780557381name_o @ ( semila1673364395name_o @ A_13 @ B_5 ) @ C_2 )
        = ( semila1673364395name_o @ A_13 @ ( semila1780557381name_o @ B_5 @ C_2 ) ) )
    <=> ( ord_less_eq_pname_o @ C_2 @ A_13 ) ) )).

thf(fact_667_Un__Int__assoc__eq,axiom,(
    ! [A_13: hoare_1167836817_state > $o,B_5: hoare_1167836817_state > $o,C_2: hoare_1167836817_state > $o] :
      ( ( ( semila1172322802tate_o @ ( semila179895820tate_o @ A_13 @ B_5 ) @ C_2 )
        = ( semila179895820tate_o @ A_13 @ ( semila1172322802tate_o @ B_5 @ C_2 ) ) )
    <=> ( ord_le827224136tate_o @ C_2 @ A_13 ) ) )).

thf(fact_668_if__image__distrib,axiom,(
    ! [P_3: pname > $o,F_5: pname > hoare_1167836817_state,G_1: pname > hoare_1167836817_state,S: pname > $o] :
      ( ( image_575578384_state
        @ ^ [X_5: pname] :
            ( if_Hoa833675553_state @ ( P_3 @ X_5 ) @ ( F_5 @ X_5 ) @ ( G_1 @ X_5 ) )
        @ S )
      = ( semila1172322802tate_o @ ( image_575578384_state @ F_5 @ ( semila1673364395name_o @ S @ ( collect_pname @ P_3 ) ) )
        @ ( image_575578384_state @ G_1
          @ ( semila1673364395name_o @ S
            @ ( collect_pname
              @ ^ [X_5: pname] :
                  ( (~) @ ( P_3 @ X_5 ) ) ) ) ) ) ) )).

thf(fact_669_dom__if,axiom,(
    ! [P_2: pname > $o,F_4: pname > option_com,G: pname > option_com] :
      ( ( dom_pname_com
        @ ^ [X_5: pname] :
            ( if_option_com @ ( P_2 @ X_5 ) @ ( F_4 @ X_5 ) @ ( G @ X_5 ) ) )
      = ( semila1780557381name_o @ ( semila1673364395name_o @ ( dom_pname_com @ F_4 ) @ ( collect_pname @ P_2 ) )
        @ ( semila1673364395name_o @ ( dom_pname_com @ G )
          @ ( collect_pname
            @ ^ [X_5: pname] :
                ( (~) @ ( P_2 @ X_5 ) ) ) ) ) ) )).

thf(fact_670_Int__Collect__mono,axiom,(
    ! [Q: pname > $o,P_1: pname > $o,A_12: pname > $o,B_4: pname > $o] :
      ( ( ord_less_eq_pname_o @ A_12 @ B_4 )
     => ( ! [X_5: pname] :
            ( ( member_pname @ X_5 @ A_12 )
           => ( ( P_1 @ X_5 )
             => ( Q @ X_5 ) ) )
       => ( ord_less_eq_pname_o @ ( semila1673364395name_o @ A_12 @ ( collect_pname @ P_1 ) ) @ ( semila1673364395name_o @ B_4 @ ( collect_pname @ Q ) ) ) ) ) )).

thf(fact_671_Int__Collect__mono,axiom,(
    ! [Q: hoare_1167836817_state > $o,P_1: hoare_1167836817_state > $o,A_12: hoare_1167836817_state > $o,B_4: hoare_1167836817_state > $o] :
      ( ( ord_le827224136tate_o @ A_12 @ B_4 )
     => ( ! [X_5: hoare_1167836817_state] :
            ( ( member2058392318_state @ X_5 @ A_12 )
           => ( ( P_1 @ X_5 )
             => ( Q @ X_5 ) ) )
       => ( ord_le827224136tate_o @ ( semila179895820tate_o @ A_12 @ ( collec1027672124_state @ P_1 ) ) @ ( semila179895820tate_o @ B_4 @ ( collec1027672124_state @ Q ) ) ) ) ) )).

thf(fact_672_Int__Collect__mono,axiom,(
    ! [Q: ( pname > $o ) > $o,P_1: ( pname > $o ) > $o,A_12: ( pname > $o ) > $o,B_4: ( pname > $o ) > $o] :
      ( ( ord_le1205211808me_o_o @ A_12 @ B_4 )
     => ( ! [X_5: pname > $o] :
            ( ( member_pname_o @ X_5 @ A_12 )
           => ( ( P_1 @ X_5 )
             => ( Q @ X_5 ) ) )
       => ( ord_le1205211808me_o_o @ ( semila2013987940me_o_o @ A_12 @ ( collect_pname_o @ P_1 ) ) @ ( semila2013987940me_o_o @ B_4 @ ( collect_pname_o @ Q ) ) ) ) ) )).

thf(fact_673_Int__Collect__mono,axiom,(
    ! [Q: ( hoare_1167836817_state > $o ) > $o,P_1: ( hoare_1167836817_state > $o ) > $o,A_12: ( hoare_1167836817_state > $o ) > $o,B_4: ( hoare_1167836817_state > $o ) > $o] :
      ( ( ord_le741939125te_o_o @ A_12 @ B_4 )
     => ( ! [X_5: hoare_1167836817_state > $o] :
            ( ( member864234961tate_o @ X_5 @ A_12 )
           => ( ( P_1 @ X_5 )
             => ( Q @ X_5 ) ) )
       => ( ord_le741939125te_o_o @ ( semila1758709489te_o_o @ A_12 @ ( collec269976083tate_o @ P_1 ) ) @ ( semila1758709489te_o_o @ B_4 @ ( collec269976083tate_o @ Q ) ) ) ) ) )).

thf(fact_674_distrib__imp1,axiom,(
    ! [X_6: hoare_1167836817_state > $o,Y_3: hoare_1167836817_state > $o,Z_2: hoare_1167836817_state > $o] :
      ( ! [X_5: hoare_1167836817_state > $o,Y_2: hoare_1167836817_state > $o,Z_1: hoare_1167836817_state > $o] :
          ( ( semila179895820tate_o @ X_5 @ ( semila1172322802tate_o @ Y_2 @ Z_1 ) )
          = ( semila1172322802tate_o @ ( semila179895820tate_o @ X_5 @ Y_2 ) @ ( semila179895820tate_o @ X_5 @ Z_1 ) ) )
     => ( ( semila1172322802tate_o @ X_6 @ ( semila179895820tate_o @ Y_3 @ Z_2 ) )
        = ( semila179895820tate_o @ ( semila1172322802tate_o @ X_6 @ Y_3 ) @ ( semila1172322802tate_o @ X_6 @ Z_2 ) ) ) ) )).

thf(fact_675_distrib__imp2,axiom,(
    ! [X_4: hoare_1167836817_state > $o,Y_1: hoare_1167836817_state > $o,Z: hoare_1167836817_state > $o] :
      ( ! [X_5: hoare_1167836817_state > $o,Y_2: hoare_1167836817_state > $o,Z_1: hoare_1167836817_state > $o] :
          ( ( semila1172322802tate_o @ X_5 @ ( semila179895820tate_o @ Y_2 @ Z_1 ) )
          = ( semila179895820tate_o @ ( semila1172322802tate_o @ X_5 @ Y_2 ) @ ( semila1172322802tate_o @ X_5 @ Z_1 ) ) )
     => ( ( semila179895820tate_o @ X_4 @ ( semila1172322802tate_o @ Y_1 @ Z ) )
        = ( semila1172322802tate_o @ ( semila179895820tate_o @ X_4 @ Y_1 ) @ ( semila179895820tate_o @ X_4 @ Z ) ) ) ) )).

thf(fact_676_folding__one_Oremove,axiom,(
    ! [X_3: pname,A_11: pname > $o,F_3: pname > pname > pname,F_2: ( pname > $o ) > pname] :
      ( ( finite1282449217_pname @ F_3 @ F_2 )
     => ( ( finite_finite_pname @ A_11 )
       => ( ( member_pname @ X_3 @ A_11 )
         => ( ( ( ( minus_minus_pname_o @ A_11 @ ( insert_pname @ X_3 @ bot_bot_pname_o ) )
                = bot_bot_pname_o )
             => ( ( F_2 @ A_11 )
                = X_3 ) )
            & ( ( ( minus_minus_pname_o @ A_11 @ ( insert_pname @ X_3 @ bot_bot_pname_o ) )
               != bot_bot_pname_o )
             => ( ( F_2 @ A_11 )
                = ( F_3 @ X_3 @ ( F_2 @ ( minus_minus_pname_o @ A_11 @ ( insert_pname @ X_3 @ bot_bot_pname_o ) ) ) ) ) ) ) ) ) ) )).

thf(fact_677_folding__one_Oremove,axiom,(
    ! [X_3: hoare_1167836817_state,A_11: hoare_1167836817_state > $o,F_3: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_2: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
      ( ( finite1074406356_state @ F_3 @ F_2 )
     => ( ( finite1084549118_state @ A_11 )
       => ( ( member2058392318_state @ X_3 @ A_11 )
         => ( ( ( ( minus_2107060239tate_o @ A_11 @ ( insert2134838167_state @ X_3 @ bot_bo70021908tate_o ) )
                = bot_bo70021908tate_o )
             => ( ( F_2 @ A_11 )
                = X_3 ) )
            & ( ( ( minus_2107060239tate_o @ A_11 @ ( insert2134838167_state @ X_3 @ bot_bo70021908tate_o ) )
               != bot_bo70021908tate_o )
             => ( ( F_2 @ A_11 )
                = ( F_3 @ X_3 @ ( F_2 @ ( minus_2107060239tate_o @ A_11 @ ( insert2134838167_state @ X_3 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ) ) ) )).

thf(fact_678_folding__one_Oremove,axiom,(
    ! [X_3: pname > $o,A_11: ( pname > $o ) > $o,F_3: ( pname > $o ) > ( pname > $o ) > pname > $o,F_2: ( ( pname > $o ) > $o ) > pname > $o] :
      ( ( finite349908348name_o @ F_3 @ F_2 )
     => ( ( finite297249702name_o @ A_11 )
       => ( ( member_pname_o @ X_3 @ A_11 )
         => ( ( ( ( minus_1480864103me_o_o @ A_11 @ ( insert_pname_o @ X_3 @ bot_bot_pname_o_o ) )
                = bot_bot_pname_o_o )
             => ( ( F_2 @ A_11 )
                = X_3 ) )
            & ( ( ( minus_1480864103me_o_o @ A_11 @ ( insert_pname_o @ X_3 @ bot_bot_pname_o_o ) )
               != bot_bot_pname_o_o )
             => ( ( F_2 @ A_11 )
                = ( F_3 @ X_3 @ ( F_2 @ ( minus_1480864103me_o_o @ A_11 @ ( insert_pname_o @ X_3 @ bot_bot_pname_o_o ) ) ) ) ) ) ) ) ) ) )).

thf(fact_679_folding__one_Oremove,axiom,(
    ! [X_3: hoare_1167836817_state > $o,A_11: ( hoare_1167836817_state > $o ) > $o,F_3: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,F_2: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o] :
      ( ( finite979047547tate_o @ F_3 @ F_2 )
     => ( ( finite1380128977tate_o @ A_11 )
       => ( ( member864234961tate_o @ X_3 @ A_11 )
         => ( ( ( ( minus_1708687022te_o_o @ A_11 @ ( insert999278200tate_o @ X_3 @ bot_bo691907561te_o_o ) )
                = bot_bo691907561te_o_o )
             => ( ( F_2 @ A_11 )
                = X_3 ) )
            & ( ( ( minus_1708687022te_o_o @ A_11 @ ( insert999278200tate_o @ X_3 @ bot_bo691907561te_o_o ) )
               != bot_bo691907561te_o_o )
             => ( ( F_2 @ A_11 )
                = ( F_3 @ X_3 @ ( F_2 @ ( minus_1708687022te_o_o @ A_11 @ ( insert999278200tate_o @ X_3 @ bot_bo691907561te_o_o ) ) ) ) ) ) ) ) ) ) )).

thf(fact_680_folding__one_Oinsert__remove,axiom,(
    ! [X_2: pname,A_10: pname > $o,F_1: pname > pname > pname,F: ( pname > $o ) > pname] :
      ( ( finite1282449217_pname @ F_1 @ F )
     => ( ( finite_finite_pname @ A_10 )
       => ( ( ( ( minus_minus_pname_o @ A_10 @ ( insert_pname @ X_2 @ bot_bot_pname_o ) )
              = bot_bot_pname_o )
           => ( ( F @ ( insert_pname @ X_2 @ A_10 ) )
              = X_2 ) )
          & ( ( ( minus_minus_pname_o @ A_10 @ ( insert_pname @ X_2 @ bot_bot_pname_o ) )
             != bot_bot_pname_o )
           => ( ( F @ ( insert_pname @ X_2 @ A_10 ) )
              = ( F_1 @ X_2 @ ( F @ ( minus_minus_pname_o @ A_10 @ ( insert_pname @ X_2 @ bot_bot_pname_o ) ) ) ) ) ) ) ) ) )).

thf(fact_681_folding__one_Oinsert__remove,axiom,(
    ! [X_2: hoare_1167836817_state,A_10: hoare_1167836817_state > $o,F_1: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
      ( ( finite1074406356_state @ F_1 @ F )
     => ( ( finite1084549118_state @ A_10 )
       => ( ( ( ( minus_2107060239tate_o @ A_10 @ ( insert2134838167_state @ X_2 @ bot_bo70021908tate_o ) )
              = bot_bo70021908tate_o )
           => ( ( F @ ( insert2134838167_state @ X_2 @ A_10 ) )
              = X_2 ) )
          & ( ( ( minus_2107060239tate_o @ A_10 @ ( insert2134838167_state @ X_2 @ bot_bo70021908tate_o ) )
             != bot_bo70021908tate_o )
           => ( ( F @ ( insert2134838167_state @ X_2 @ A_10 ) )
              = ( F_1 @ X_2 @ ( F @ ( minus_2107060239tate_o @ A_10 @ ( insert2134838167_state @ X_2 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ) ) )).

thf(fact_682_folding__one_Oinsert__remove,axiom,(
    ! [X_2: pname > $o,A_10: ( pname > $o ) > $o,F_1: ( pname > $o ) > ( pname > $o ) > pname > $o,F: ( ( pname > $o ) > $o ) > pname > $o] :
      ( ( finite349908348name_o @ F_1 @ F )
     => ( ( finite297249702name_o @ A_10 )
       => ( ( ( ( minus_1480864103me_o_o @ A_10 @ ( insert_pname_o @ X_2 @ bot_bot_pname_o_o ) )
              = bot_bot_pname_o_o )
           => ( ( F @ ( insert_pname_o @ X_2 @ A_10 ) )
              = X_2 ) )
          & ( ( ( minus_1480864103me_o_o @ A_10 @ ( insert_pname_o @ X_2 @ bot_bot_pname_o_o ) )
             != bot_bot_pname_o_o )
           => ( ( F @ ( insert_pname_o @ X_2 @ A_10 ) )
              = ( F_1 @ X_2 @ ( F @ ( minus_1480864103me_o_o @ A_10 @ ( insert_pname_o @ X_2 @ bot_bot_pname_o_o ) ) ) ) ) ) ) ) ) )).

thf(fact_683_folding__one_Oinsert__remove,axiom,(
    ! [X_2: hoare_1167836817_state > $o,A_10: ( hoare_1167836817_state > $o ) > $o,F_1: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,F: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o] :
      ( ( finite979047547tate_o @ F_1 @ F )
     => ( ( finite1380128977tate_o @ A_10 )
       => ( ( ( ( minus_1708687022te_o_o @ A_10 @ ( insert999278200tate_o @ X_2 @ bot_bo691907561te_o_o ) )
              = bot_bo691907561te_o_o )
           => ( ( F @ ( insert999278200tate_o @ X_2 @ A_10 ) )
              = X_2 ) )
          & ( ( ( minus_1708687022te_o_o @ A_10 @ ( insert999278200tate_o @ X_2 @ bot_bo691907561te_o_o ) )
             != bot_bo691907561te_o_o )
           => ( ( F @ ( insert999278200tate_o @ X_2 @ A_10 ) )
              = ( F_1 @ X_2 @ ( F @ ( minus_1708687022te_o_o @ A_10 @ ( insert999278200tate_o @ X_2 @ bot_bo691907561te_o_o ) ) ) ) ) ) ) ) ) )).

thf(fact_684_DiffI,axiom,(
    ! [B_3: pname > $o,C_1: pname,A_9: pname > $o] :
      ( ( member_pname @ C_1 @ A_9 )
     => ( ~ ( member_pname @ C_1 @ B_3 )
       => ( member_pname @ C_1 @ ( minus_minus_pname_o @ A_9 @ B_3 ) ) ) ) )).

thf(fact_685_DiffI,axiom,(
    ! [B_3: hoare_1167836817_state > $o,C_1: hoare_1167836817_state,A_9: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ C_1 @ A_9 )
     => ( ~ ( member2058392318_state @ C_1 @ B_3 )
       => ( member2058392318_state @ C_1 @ ( minus_2107060239tate_o @ A_9 @ B_3 ) ) ) ) )).

thf(fact_686_DiffE,axiom,(
    ! [C: pname,A_8: pname > $o,B_2: pname > $o] :
      ( ( member_pname @ C @ ( minus_minus_pname_o @ A_8 @ B_2 ) )
     => ~ ( ( member_pname @ C @ A_8 )
         => ( member_pname @ C @ B_2 ) ) ) )).

thf(fact_687_DiffE,axiom,(
    ! [C: hoare_1167836817_state,A_8: hoare_1167836817_state > $o,B_2: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ C @ ( minus_2107060239tate_o @ A_8 @ B_2 ) )
     => ~ ( ( member2058392318_state @ C @ A_8 )
         => ( member2058392318_state @ C @ B_2 ) ) ) )).

thf(fact_688_finite__Diff,axiom,(
    ! [B_1: ( pname > $o ) > $o,A_7: ( pname > $o ) > $o] :
      ( ( finite297249702name_o @ A_7 )
     => ( finite297249702name_o @ ( minus_1480864103me_o_o @ A_7 @ B_1 ) ) ) )).

thf(fact_689_finite__Diff,axiom,(
    ! [B_1: ( hoare_1167836817_state > $o ) > $o,A_7: ( hoare_1167836817_state > $o ) > $o] :
      ( ( finite1380128977tate_o @ A_7 )
     => ( finite1380128977tate_o @ ( minus_1708687022te_o_o @ A_7 @ B_1 ) ) ) )).

thf(fact_690_finite__Diff,axiom,(
    ! [B_1: pname > $o,A_7: pname > $o] :
      ( ( finite_finite_pname @ A_7 )
     => ( finite_finite_pname @ ( minus_minus_pname_o @ A_7 @ B_1 ) ) ) )).

thf(fact_691_finite__Diff,axiom,(
    ! [B_1: hoare_1167836817_state > $o,A_7: hoare_1167836817_state > $o] :
      ( ( finite1084549118_state @ A_7 )
     => ( finite1084549118_state @ ( minus_2107060239tate_o @ A_7 @ B_1 ) ) ) )).

thf(fact_692_insert__Diff,axiom,(
    ! [A_6: pname,A_5: pname > $o] :
      ( ( member_pname @ A_6 @ A_5 )
     => ( ( insert_pname @ A_6 @ ( minus_minus_pname_o @ A_5 @ ( insert_pname @ A_6 @ bot_bot_pname_o ) ) )
        = A_5 ) ) )).

thf(fact_693_insert__Diff,axiom,(
    ! [A_6: hoare_1167836817_state,A_5: hoare_1167836817_state > $o] :
      ( ( member2058392318_state @ A_6 @ A_5 )
     => ( ( insert2134838167_state @ A_6 @ ( minus_2107060239tate_o @ A_5 @ ( insert2134838167_state @ A_6 @ bot_bo70021908tate_o ) ) )
        = A_5 ) ) )).

thf(fact_694_Diff__insert__absorb,axiom,(
    ! [X_1: pname,A_4: pname > $o] :
      ( ~ ( member_pname @ X_1 @ A_4 )
     => ( ( minus_minus_pname_o @ ( insert_pname @ X_1 @ A_4 ) @ ( insert_pname @ X_1 @ bot_bot_pname_o ) )
        = A_4 ) ) )).

thf(fact_695_Diff__insert__absorb,axiom,(
    ! [X_1: hoare_1167836817_state,A_4: hoare_1167836817_state > $o] :
      ( ~ ( member2058392318_state @ X_1 @ A_4 )
     => ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_1 @ A_4 ) @ ( insert2134838167_state @ X_1 @ bot_bo70021908tate_o ) )
        = A_4 ) ) )).

thf(fact_696_insert__Diff__single,axiom,(
    ! [A_3: pname,A_2: pname > $o] :
      ( ( insert_pname @ A_3 @ ( minus_minus_pname_o @ A_2 @ ( insert_pname @ A_3 @ bot_bot_pname_o ) ) )
      = ( insert_pname @ A_3 @ A_2 ) ) )).

thf(fact_697_insert__Diff__single,axiom,(
    ! [A_3: hoare_1167836817_state,A_2: hoare_1167836817_state > $o] :
      ( ( insert2134838167_state @ A_3 @ ( minus_2107060239tate_o @ A_2 @ ( insert2134838167_state @ A_3 @ bot_bo70021908tate_o ) ) )
      = ( insert2134838167_state @ A_3 @ A_2 ) ) )).

thf(fact_698_Diff__insert2,axiom,(
    ! [A_1: hoare_1167836817_state > $o,A: hoare_1167836817_state,B: hoare_1167836817_state > $o] :
      ( ( minus_2107060239tate_o @ A_1 @ ( insert2134838167_state @ A @ B ) )
      = ( minus_2107060239tate_o @ ( minus_2107060239tate_o @ A_1 @ ( insert2134838167_state @ A @ bot_bo70021908tate_o ) ) @ B ) ) )).

%----Helper facts (16)
thf(help_fequal_1_1_fequal_000tc__Com__Opname_T,axiom,(
    ! [X: pname,Y: pname] :
      ( ~ ( fequal_pname @ X @ Y )
      | ( X = Y ) ) )).

thf(help_fequal_2_1_fequal_000tc__Com__Opname_T,axiom,(
    ! [X: pname,Y: pname] :
      ( ( X != Y )
      | ( fequal_pname @ X @ Y ) ) )).

thf(help_fequal_1_1_fequal_000tc__Com__Ostate_T,axiom,(
    ! [X: state,Y: state] :
      ( ~ ( fequal_state @ X @ Y )
      | ( X = Y ) ) )).

thf(help_fequal_2_1_fequal_000tc__Com__Ostate_T,axiom,(
    ! [X: state,Y: state] :
      ( ( X != Y )
      | ( fequal_state @ X @ Y ) ) )).

thf(help_fequal_1_1_fequal_000_062_Itc__Com__Opname_M_Eo_J_T,axiom,(
    ! [X: pname > $o,Y: pname > $o] :
      ( ~ ( fequal_pname_o @ X @ Y )
      | ( X = Y ) ) )).

thf(help_fequal_2_1_fequal_000_062_Itc__Com__Opname_M_Eo_J_T,axiom,(
    ! [X: pname > $o,Y: pname > $o] :
      ( ( X != Y )
      | ( fequal_pname_o @ X @ Y ) ) )).

thf(help_If_1_1_If_000tc__Option__Ooption_Itc__Com__Ocom_J_T,axiom,(
    ! [X: option_com,Y: option_com] :
      ( ( if_option_com @ $true @ X @ Y )
      = X ) )).

thf(help_If_2_1_If_000tc__Option__Ooption_Itc__Com__Ocom_J_T,axiom,(
    ! [X: option_com,Y: option_com] :
      ( ( if_option_com @ $false @ X @ Y )
      = Y ) )).

thf(help_If_3_1_If_000tc__Option__Ooption_Itc__Com__Ocom_J_T,axiom,(
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) )).

thf(help_If_1_1_If_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate,axiom,(
    ! [X: hoare_1167836817_state,Y: hoare_1167836817_state] :
      ( ( if_Hoa833675553_state @ $true @ X @ Y )
      = X ) )).

thf(help_If_2_1_If_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate,axiom,(
    ! [X: hoare_1167836817_state,Y: hoare_1167836817_state] :
      ( ( if_Hoa833675553_state @ $false @ X @ Y )
      = Y ) )).

thf(help_If_3_1_If_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate,axiom,(
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) )).

thf(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,axiom,(
    ! [X: hoare_1167836817_state,Y: hoare_1167836817_state] :
      ( ~ ( fequal1831255762_state @ X @ Y )
      | ( X = Y ) ) )).

thf(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,axiom,(
    ! [X: hoare_1167836817_state,Y: hoare_1167836817_state] :
      ( ( X != Y )
      | ( fequal1831255762_state @ X @ Y ) ) )).

thf(help_fequal_1_1_fequal_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_It,axiom,(
    ! [X: hoare_1167836817_state > $o,Y: hoare_1167836817_state > $o] :
      ( ~ ( fequal1486222077tate_o @ X @ Y )
      | ( X = Y ) ) )).

thf(help_fequal_2_1_fequal_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_It,axiom,(
    ! [X: hoare_1167836817_state > $o,Y: hoare_1167836817_state > $o] :
      ( ( X != Y )
      | ( fequal1486222077tate_o @ X @ Y ) ) )).

%----Conjectures (8)
thf(conj_0,hypothesis,(
    hoare_1201148605gleton )).

thf(conj_1,hypothesis,(
    wT_bodies )).

thf(conj_2,hypothesis,
    ( finite1084549118_state @ fa )).

thf(conj_3,hypothesis,(
    ~ ( member2058392318_state @ ( hoare_Mirabelle_MGT @ y ) @ fa ) )).

thf(conj_4,hypothesis,
    ( ord_le827224136tate_o @ fa
    @ ( image_575578384_state
      @ ^ [Pn: pname] :
          ( hoare_Mirabelle_MGT @ ( the_com @ ( body @ Pn ) ) )
      @ ( dom_pname_com @ body ) ) )).

thf(conj_5,hypothesis,
    ( ( body @ pn )
    = ( some_com @ y ) )).

thf(conj_6,hypothesis,
    ( hoare_123228589_state
    @ ( image_575578384_state
      @ ^ [Pn: pname] :
          ( hoare_Mirabelle_MGT @ ( body_1 @ Pn ) )
      @ ( dom_pname_com @ body ) )
    @ fa )).

thf(conj_7,conjecture,
    ( hoare_123228589_state
    @ ( image_575578384_state
      @ ^ [Pn: pname] :
          ( hoare_Mirabelle_MGT @ ( body_1 @ Pn ) )
      @ ( dom_pname_com @ body ) )
    @ ( insert2134838167_state @ ( hoare_Mirabelle_MGT @ y ) @ bot_bo70021908tate_o ) )).

%------------------------------------------------------------------------------
